Graphics.Rendering.Chart.Axis.Types:invLinMap from Chart-1.5.3

Percentage Accurate: 67.5% → 86.2%

Time: 9.8s

Alternatives: 22

Speedup: 0.8×

Specification

Math

\[\begin{array}{l} \\ x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (+ x (/ (* (- y z) (- t x)) (- a z))))

C

double code(double x, double y, double z, double t, double a) {
	return x + (((y - z) * (t - x)) / (a - z));
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x + (((y - z) * (t - x)) / (a - z))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return x + (((y - z) * (t - x)) / (a - z));
}

Python

def code(x, y, z, t, a):
	return x + (((y - z) * (t - x)) / (a - z))

Julia

function code(x, y, z, t, a)
	return Float64(x + Float64(Float64(Float64(y - z) * Float64(t - x)) / Float64(a - z)))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = x + (((y - z) * (t - x)) / (a - z));
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(x + N[(N[(N[(y - z), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 67.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (+ x (/ (* (- y z) (- t x)) (- a z))))

C

double code(double x, double y, double z, double t, double a) {
	return x + (((y - z) * (t - x)) / (a - z));
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x + (((y - z) * (t - x)) / (a - z))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return x + (((y - z) * (t - x)) / (a - z));
}

Python

def code(x, y, z, t, a):
	return x + (((y - z) * (t - x)) / (a - z))

Julia

function code(x, y, z, t, a)
	return Float64(x + Float64(Float64(Float64(y - z) * Float64(t - x)) / Float64(a - z)))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = x + (((y - z) * (t - x)) / (a - z));
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(x + N[(N[(N[(y - z), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 86.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{-\left(t - x\right)}{z}, y - a, t\right)\\ t_2 := x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-204}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{-308}:\\ \;\;\;\;t - \left(-x\right) \cdot \frac{y - a}{z}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+307}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- (- t x)) z) (- y a) t))
        (t_2 (+ x (/ (* (- y z) (- t x)) (- a z)))))
   (if (<= t_2 (- INFINITY))
     t_1
     (if (<= t_2 -5e-204)
       t_2
       (if (<= t_2 4e-308)
         (- t (* (- x) (/ (- y a) z)))
         (if (<= t_2 2e+307) t_2 t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((-(t - x) / z), (y - a), t);
	double t_2 = x + (((y - z) * (t - x)) / (a - z));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= -5e-204) {
		tmp = t_2;
	} else if (t_2 <= 4e-308) {
		tmp = t - (-x * ((y - a) / z));
	} else if (t_2 <= 2e+307) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(-Float64(t - x)) / z), Float64(y - a), t)
	t_2 = Float64(x + Float64(Float64(Float64(y - z) * Float64(t - x)) / Float64(a - z)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= -5e-204)
		tmp = t_2;
	elseif (t_2 <= 4e-308)
		tmp = Float64(t - Float64(Float64(-x) * Float64(Float64(y - a) / z)));
	elseif (t_2 <= 2e+307)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[((-N[(t - x), $MachinePrecision]) / z), $MachinePrecision] * N[(y - a), $MachinePrecision] + t), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(N[(y - z), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, -5e-204], t$95$2, If[LessEqual[t$95$2, 4e-308], N[(t - N[((-x) * N[(N[(y - a), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+307], t$95$2, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < -inf.0 or 1.99999999999999997e307 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z)))

   1. Initial program 42.7%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{y \cdot \left(t - x\right)}{z}\right) - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{t + \left(-1 \cdot \frac{y \cdot \left(t - x\right)}{z} - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      2. distribute-lft-out-- N/A

         \[\leadsto t + \color{blue}{-1 \cdot \left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      3. div-sub N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + t} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right)} + t \]

      6. div-sub N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)}\right)\right) + t \]

      7. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\left(\color{blue}{y \cdot \frac{t - x}{z}} - \frac{a \cdot \left(t - x\right)}{z}\right)\right)\right) + t \]

      8. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\left(y \cdot \frac{t - x}{z} - \color{blue}{a \cdot \frac{t - x}{z}}\right)\right)\right) + t \]

      9. distribute-rgt-out-- N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{t - x}{z} \cdot \left(y - a\right)}\right)\right) + t \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{t - x}{z}\right)\right) \cdot \left(y - a\right)} + t \]

      11. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\frac{t - x}{z}\right), y - a, t\right)} \]

   5. Applied rewrites 82.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-\left(t - x\right)}{z}, y - a, t\right)} \]

   if -inf.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < -5.0000000000000002e-204 or 4.00000000000000013e-308 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < 1.99999999999999997e307

   1. Initial program 97.2%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   if -5.0000000000000002e-204 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < 4.00000000000000013e-308

   1. Initial program 4.5%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(y - z\right) \cdot \left(t - x\right)}}{a - z} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t - x}{a - z} \cdot \left(y - z\right)} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

      8. lower-/.f64 5.2

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{t - x}{a - z}}, y - z, x\right) \]

   4. Applied rewrites 5.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{y \cdot \left(t - x\right)}{z}\right) - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}} \]
   6. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{y \cdot \left(t - x\right)}{z}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(t - x\right)}{z}} \]

      2. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(t - \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{y \cdot \left(t - x\right)}{z}\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(t - x\right)}{z} \]

      3. metadata-eval N/A

         \[\leadsto \left(t - \color{blue}{1} \cdot \frac{y \cdot \left(t - x\right)}{z}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(t - x\right)}{z} \]

      4. *-lft-identity N/A

         \[\leadsto \left(t - \color{blue}{\frac{y \cdot \left(t - x\right)}{z}}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(t - x\right)}{z} \]

      5. metadata-eval N/A

         \[\leadsto \left(t - \frac{y \cdot \left(t - x\right)}{z}\right) + \color{blue}{1} \cdot \frac{a \cdot \left(t - x\right)}{z} \]

      6. *-lft-identity N/A

         \[\leadsto \left(t - \frac{y \cdot \left(t - x\right)}{z}\right) + \color{blue}{\frac{a \cdot \left(t - x\right)}{z}} \]

      7. associate-+l- N/A

         \[\leadsto \color{blue}{t - \left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      8. div-sub N/A

         \[\leadsto t - \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      9. lower--.f64 N/A

         \[\leadsto \color{blue}{t - \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      10. div-sub N/A

          \[\leadsto t - \color{blue}{\left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      11. associate-/l* N/A

          \[\leadsto t - \left(\color{blue}{y \cdot \frac{t - x}{z}} - \frac{a \cdot \left(t - x\right)}{z}\right) \]

      12. associate-/l* N/A

          \[\leadsto t - \left(y \cdot \frac{t - x}{z} - \color{blue}{a \cdot \frac{t - x}{z}}\right) \]

      13. distribute-rgt-out-- N/A

          \[\leadsto t - \color{blue}{\frac{t - x}{z} \cdot \left(y - a\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto t - \color{blue}{\frac{t - x}{z} \cdot \left(y - a\right)} \]

      15. lower-/.f64 N/A

          \[\leadsto t - \color{blue}{\frac{t - x}{z}} \cdot \left(y - a\right) \]

      16. lower--.f64 N/A

          \[\leadsto t - \frac{\color{blue}{t - x}}{z} \cdot \left(y - a\right) \]

      17. lower--.f64 92.2

          \[\leadsto t - \frac{t - x}{z} \cdot \color{blue}{\left(y - a\right)} \]

   7. Applied rewrites 92.2%

      \[\leadsto \color{blue}{t - \frac{t - x}{z} \cdot \left(y - a\right)} \]

   8. Taylor expanded in x around inf

      \[\leadsto t - -1 \cdot \color{blue}{\frac{x \cdot \left(y - a\right)}{z}} \]
   9. Step-by-step derivation

   10. Applied rewrites 99.8%

       \[\leadsto t - \left(-x\right) \cdot \color{blue}{\frac{y - a}{z}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{-\left(t - x\right)}{z}, y - a, t\right)\\ \mathbf{elif}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \leq -5 \cdot 10^{-204}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\\ \mathbf{elif}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \leq 4 \cdot 10^{-308}:\\ \;\;\;\;t - \left(-x\right) \cdot \frac{y - a}{z}\\ \mathbf{elif}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \leq 2 \cdot 10^{+307}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-\left(t - x\right)}{z}, y - a, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{-\left(t - x\right)}{z}, y - a, t\right)\\ t_2 := x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\\ t_3 := x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+305}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-204}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{-308}:\\ \;\;\;\;t - \left(-x\right) \cdot \frac{y - a}{z}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+307}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- (- t x)) z) (- y a) t))
        (t_2 (+ x (/ (* (- y z) (- t x)) (- a z))))
        (t_3 (+ x (/ (* (- y z) t) (- a z)))))
   (if (<= t_2 -5e+305)
     t_1
     (if (<= t_2 -5e-204)
       t_3
       (if (<= t_2 4e-308)
         (- t (* (- x) (/ (- y a) z)))
         (if (<= t_2 2e+307) t_3 t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((-(t - x) / z), (y - a), t);
	double t_2 = x + (((y - z) * (t - x)) / (a - z));
	double t_3 = x + (((y - z) * t) / (a - z));
	double tmp;
	if (t_2 <= -5e+305) {
		tmp = t_1;
	} else if (t_2 <= -5e-204) {
		tmp = t_3;
	} else if (t_2 <= 4e-308) {
		tmp = t - (-x * ((y - a) / z));
	} else if (t_2 <= 2e+307) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(-Float64(t - x)) / z), Float64(y - a), t)
	t_2 = Float64(x + Float64(Float64(Float64(y - z) * Float64(t - x)) / Float64(a - z)))
	t_3 = Float64(x + Float64(Float64(Float64(y - z) * t) / Float64(a - z)))
	tmp = 0.0
	if (t_2 <= -5e+305)
		tmp = t_1;
	elseif (t_2 <= -5e-204)
		tmp = t_3;
	elseif (t_2 <= 4e-308)
		tmp = Float64(t - Float64(Float64(-x) * Float64(Float64(y - a) / z)));
	elseif (t_2 <= 2e+307)
		tmp = t_3;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[((-N[(t - x), $MachinePrecision]) / z), $MachinePrecision] * N[(y - a), $MachinePrecision] + t), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(N[(y - z), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(x + N[(N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+305], t$95$1, If[LessEqual[t$95$2, -5e-204], t$95$3, If[LessEqual[t$95$2, 4e-308], N[(t - N[((-x) * N[(N[(y - a), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+307], t$95$3, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < -5.00000000000000009e305 or 1.99999999999999997e307 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z)))

   1. Initial program 43.3%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{y \cdot \left(t - x\right)}{z}\right) - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{t + \left(-1 \cdot \frac{y \cdot \left(t - x\right)}{z} - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      2. distribute-lft-out-- N/A

         \[\leadsto t + \color{blue}{-1 \cdot \left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      3. div-sub N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + t} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right)} + t \]

      6. div-sub N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)}\right)\right) + t \]

      7. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\left(\color{blue}{y \cdot \frac{t - x}{z}} - \frac{a \cdot \left(t - x\right)}{z}\right)\right)\right) + t \]

      8. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\left(y \cdot \frac{t - x}{z} - \color{blue}{a \cdot \frac{t - x}{z}}\right)\right)\right) + t \]

      9. distribute-rgt-out-- N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{t - x}{z} \cdot \left(y - a\right)}\right)\right) + t \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{t - x}{z}\right)\right) \cdot \left(y - a\right)} + t \]

      11. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\frac{t - x}{z}\right), y - a, t\right)} \]

   5. Applied rewrites 81.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-\left(t - x\right)}{z}, y - a, t\right)} \]

   if -5.00000000000000009e305 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < -5.0000000000000002e-204 or 4.00000000000000013e-308 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < 1.99999999999999997e307

   1. Initial program 97.2%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto x + \frac{\color{blue}{t \cdot \left(y - z\right)}}{a - z} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x + \frac{\color{blue}{\left(y - z\right) \cdot t}}{a - z} \]

      2. lower-*.f64 N/A

         \[\leadsto x + \frac{\color{blue}{\left(y - z\right) \cdot t}}{a - z} \]

      3. lower--.f64 83.1

         \[\leadsto x + \frac{\color{blue}{\left(y - z\right)} \cdot t}{a - z} \]

   5. Applied rewrites 83.1%

      \[\leadsto x + \frac{\color{blue}{\left(y - z\right) \cdot t}}{a - z} \]

   if -5.0000000000000002e-204 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < 4.00000000000000013e-308

   1. Initial program 4.5%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(y - z\right) \cdot \left(t - x\right)}}{a - z} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t - x}{a - z} \cdot \left(y - z\right)} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

      8. lower-/.f64 5.2

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{t - x}{a - z}}, y - z, x\right) \]

   4. Applied rewrites 5.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{y \cdot \left(t - x\right)}{z}\right) - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}} \]
   6. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{y \cdot \left(t - x\right)}{z}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(t - x\right)}{z}} \]

      2. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(t - \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{y \cdot \left(t - x\right)}{z}\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(t - x\right)}{z} \]

      3. metadata-eval N/A

         \[\leadsto \left(t - \color{blue}{1} \cdot \frac{y \cdot \left(t - x\right)}{z}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(t - x\right)}{z} \]

      4. *-lft-identity N/A

         \[\leadsto \left(t - \color{blue}{\frac{y \cdot \left(t - x\right)}{z}}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(t - x\right)}{z} \]

      5. metadata-eval N/A

         \[\leadsto \left(t - \frac{y \cdot \left(t - x\right)}{z}\right) + \color{blue}{1} \cdot \frac{a \cdot \left(t - x\right)}{z} \]

      6. *-lft-identity N/A

         \[\leadsto \left(t - \frac{y \cdot \left(t - x\right)}{z}\right) + \color{blue}{\frac{a \cdot \left(t - x\right)}{z}} \]

      7. associate-+l- N/A

         \[\leadsto \color{blue}{t - \left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      8. div-sub N/A

         \[\leadsto t - \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      9. lower--.f64 N/A

         \[\leadsto \color{blue}{t - \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      10. div-sub N/A

          \[\leadsto t - \color{blue}{\left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      11. associate-/l* N/A

          \[\leadsto t - \left(\color{blue}{y \cdot \frac{t - x}{z}} - \frac{a \cdot \left(t - x\right)}{z}\right) \]

      12. associate-/l* N/A

          \[\leadsto t - \left(y \cdot \frac{t - x}{z} - \color{blue}{a \cdot \frac{t - x}{z}}\right) \]

      13. distribute-rgt-out-- N/A

          \[\leadsto t - \color{blue}{\frac{t - x}{z} \cdot \left(y - a\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto t - \color{blue}{\frac{t - x}{z} \cdot \left(y - a\right)} \]

      15. lower-/.f64 N/A

          \[\leadsto t - \color{blue}{\frac{t - x}{z}} \cdot \left(y - a\right) \]

      16. lower--.f64 N/A

          \[\leadsto t - \frac{\color{blue}{t - x}}{z} \cdot \left(y - a\right) \]

      17. lower--.f64 92.2

          \[\leadsto t - \frac{t - x}{z} \cdot \color{blue}{\left(y - a\right)} \]

   7. Applied rewrites 92.2%

      \[\leadsto \color{blue}{t - \frac{t - x}{z} \cdot \left(y - a\right)} \]

   8. Taylor expanded in x around inf

      \[\leadsto t - -1 \cdot \color{blue}{\frac{x \cdot \left(y - a\right)}{z}} \]
   9. Step-by-step derivation

   10. Applied rewrites 99.8%

       \[\leadsto t - \left(-x\right) \cdot \color{blue}{\frac{y - a}{z}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \leq -5 \cdot 10^{+305}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-\left(t - x\right)}{z}, y - a, t\right)\\ \mathbf{elif}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \leq -5 \cdot 10^{-204}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{elif}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \leq 4 \cdot 10^{-308}:\\ \;\;\;\;t - \left(-x\right) \cdot \frac{y - a}{z}\\ \mathbf{elif}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \leq 2 \cdot 10^{+307}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-\left(t - x\right)}{z}, y - a, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)\\ t_2 := x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-204}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{-308}:\\ \;\;\;\;t - \left(-x\right) \cdot \frac{y - a}{z}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-64}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- t x) (- a z)) (- y z) x))
        (t_2 (+ x (/ (* (- y z) (- t x)) (- a z)))))
   (if (<= t_2 -5e-204)
     t_1
     (if (<= t_2 4e-308)
       (- t (* (- x) (/ (- y a) z)))
       (if (<= t_2 5e-64) (+ x (/ (* (- y z) t) (- a z))) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((t - x) / (a - z)), (y - z), x);
	double t_2 = x + (((y - z) * (t - x)) / (a - z));
	double tmp;
	if (t_2 <= -5e-204) {
		tmp = t_1;
	} else if (t_2 <= 4e-308) {
		tmp = t - (-x * ((y - a) / z));
	} else if (t_2 <= 5e-64) {
		tmp = x + (((y - z) * t) / (a - z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(t - x) / Float64(a - z)), Float64(y - z), x)
	t_2 = Float64(x + Float64(Float64(Float64(y - z) * Float64(t - x)) / Float64(a - z)))
	tmp = 0.0
	if (t_2 <= -5e-204)
		tmp = t_1;
	elseif (t_2 <= 4e-308)
		tmp = Float64(t - Float64(Float64(-x) * Float64(Float64(y - a) / z)));
	elseif (t_2 <= 5e-64)
		tmp = Float64(x + Float64(Float64(Float64(y - z) * t) / Float64(a - z)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(N[(y - z), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e-204], t$95$1, If[LessEqual[t$95$2, 4e-308], N[(t - N[((-x) * N[(N[(y - a), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e-64], N[(x + N[(N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < -5.0000000000000002e-204 or 5.00000000000000033e-64 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z)))

   1. Initial program 73.8%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(y - z\right) \cdot \left(t - x\right)}}{a - z} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t - x}{a - z} \cdot \left(y - z\right)} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

      8. lower-/.f64 85.9

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{t - x}{a - z}}, y - z, x\right) \]

   4. Applied rewrites 85.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

   if -5.0000000000000002e-204 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < 4.00000000000000013e-308

   1. Initial program 4.5%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(y - z\right) \cdot \left(t - x\right)}}{a - z} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t - x}{a - z} \cdot \left(y - z\right)} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

      8. lower-/.f64 5.2

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{t - x}{a - z}}, y - z, x\right) \]

   4. Applied rewrites 5.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{y \cdot \left(t - x\right)}{z}\right) - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}} \]
   6. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{y \cdot \left(t - x\right)}{z}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(t - x\right)}{z}} \]

      2. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(t - \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{y \cdot \left(t - x\right)}{z}\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(t - x\right)}{z} \]

      3. metadata-eval N/A

         \[\leadsto \left(t - \color{blue}{1} \cdot \frac{y \cdot \left(t - x\right)}{z}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(t - x\right)}{z} \]

      4. *-lft-identity N/A

         \[\leadsto \left(t - \color{blue}{\frac{y \cdot \left(t - x\right)}{z}}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(t - x\right)}{z} \]

      5. metadata-eval N/A

         \[\leadsto \left(t - \frac{y \cdot \left(t - x\right)}{z}\right) + \color{blue}{1} \cdot \frac{a \cdot \left(t - x\right)}{z} \]

      6. *-lft-identity N/A

         \[\leadsto \left(t - \frac{y \cdot \left(t - x\right)}{z}\right) + \color{blue}{\frac{a \cdot \left(t - x\right)}{z}} \]

      7. associate-+l- N/A

         \[\leadsto \color{blue}{t - \left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      8. div-sub N/A

         \[\leadsto t - \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      9. lower--.f64 N/A

         \[\leadsto \color{blue}{t - \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      10. div-sub N/A

          \[\leadsto t - \color{blue}{\left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      11. associate-/l* N/A

          \[\leadsto t - \left(\color{blue}{y \cdot \frac{t - x}{z}} - \frac{a \cdot \left(t - x\right)}{z}\right) \]

      12. associate-/l* N/A

          \[\leadsto t - \left(y \cdot \frac{t - x}{z} - \color{blue}{a \cdot \frac{t - x}{z}}\right) \]

      13. distribute-rgt-out-- N/A

          \[\leadsto t - \color{blue}{\frac{t - x}{z} \cdot \left(y - a\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto t - \color{blue}{\frac{t - x}{z} \cdot \left(y - a\right)} \]

      15. lower-/.f64 N/A

          \[\leadsto t - \color{blue}{\frac{t - x}{z}} \cdot \left(y - a\right) \]

      16. lower--.f64 N/A

          \[\leadsto t - \frac{\color{blue}{t - x}}{z} \cdot \left(y - a\right) \]

      17. lower--.f64 92.2

          \[\leadsto t - \frac{t - x}{z} \cdot \color{blue}{\left(y - a\right)} \]

   7. Applied rewrites 92.2%

      \[\leadsto \color{blue}{t - \frac{t - x}{z} \cdot \left(y - a\right)} \]

   8. Taylor expanded in x around inf

      \[\leadsto t - -1 \cdot \color{blue}{\frac{x \cdot \left(y - a\right)}{z}} \]
   9. Step-by-step derivation

   10. Applied rewrites 99.8%

       \[\leadsto t - \left(-x\right) \cdot \color{blue}{\frac{y - a}{z}} \]

   if 4.00000000000000013e-308 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < 5.00000000000000033e-64

   1. Initial program 99.6%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto x + \frac{\color{blue}{t \cdot \left(y - z\right)}}{a - z} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x + \frac{\color{blue}{\left(y - z\right) \cdot t}}{a - z} \]

      2. lower-*.f64 N/A

         \[\leadsto x + \frac{\color{blue}{\left(y - z\right) \cdot t}}{a - z} \]

      3. lower--.f64 99.6

         \[\leadsto x + \frac{\color{blue}{\left(y - z\right)} \cdot t}{a - z} \]

   5. Applied rewrites 99.6%

      \[\leadsto x + \frac{\color{blue}{\left(y - z\right) \cdot t}}{a - z} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 77.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{-\left(t - x\right)}{z}, y - a, t\right)\\ \mathbf{if}\;z \leq -4.6 \cdot 10^{+20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-12}:\\ \;\;\;\;x + \frac{\left(t - x\right) \cdot y}{a - z}\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+31}:\\ \;\;\;\;x + \frac{y - z}{a} \cdot \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- (- t x)) z) (- y a) t)))
   (if (<= z -4.6e+20)
     t_1
     (if (<= z 2.3e-12)
       (+ x (/ (* (- t x) y) (- a z)))
       (if (<= z 2.5e+31) (+ x (* (/ (- y z) a) (- t x))) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((-(t - x) / z), (y - a), t);
	double tmp;
	if (z <= -4.6e+20) {
		tmp = t_1;
	} else if (z <= 2.3e-12) {
		tmp = x + (((t - x) * y) / (a - z));
	} else if (z <= 2.5e+31) {
		tmp = x + (((y - z) / a) * (t - x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(-Float64(t - x)) / z), Float64(y - a), t)
	tmp = 0.0
	if (z <= -4.6e+20)
		tmp = t_1;
	elseif (z <= 2.3e-12)
		tmp = Float64(x + Float64(Float64(Float64(t - x) * y) / Float64(a - z)));
	elseif (z <= 2.5e+31)
		tmp = Float64(x + Float64(Float64(Float64(y - z) / a) * Float64(t - x)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[((-N[(t - x), $MachinePrecision]) / z), $MachinePrecision] * N[(y - a), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -4.6e+20], t$95$1, If[LessEqual[z, 2.3e-12], N[(x + N[(N[(N[(t - x), $MachinePrecision] * y), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.5e+31], N[(x + N[(N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.6e20 or 2.50000000000000013e31 < z

   1. Initial program 43.6%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{y \cdot \left(t - x\right)}{z}\right) - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{t + \left(-1 \cdot \frac{y \cdot \left(t - x\right)}{z} - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      2. distribute-lft-out-- N/A

         \[\leadsto t + \color{blue}{-1 \cdot \left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      3. div-sub N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + t} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right)} + t \]

      6. div-sub N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)}\right)\right) + t \]

      7. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\left(\color{blue}{y \cdot \frac{t - x}{z}} - \frac{a \cdot \left(t - x\right)}{z}\right)\right)\right) + t \]

      8. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\left(y \cdot \frac{t - x}{z} - \color{blue}{a \cdot \frac{t - x}{z}}\right)\right)\right) + t \]

      9. distribute-rgt-out-- N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{t - x}{z} \cdot \left(y - a\right)}\right)\right) + t \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{t - x}{z}\right)\right) \cdot \left(y - a\right)} + t \]

      11. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\frac{t - x}{z}\right), y - a, t\right)} \]

   5. Applied rewrites 81.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-\left(t - x\right)}{z}, y - a, t\right)} \]

   if -4.6e20 < z < 2.29999999999999989e-12

   1. Initial program 91.9%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto x + \frac{\color{blue}{y \cdot \left(t - x\right)}}{a - z} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x + \frac{\color{blue}{\left(t - x\right) \cdot y}}{a - z} \]

      2. lower-*.f64 N/A

         \[\leadsto x + \frac{\color{blue}{\left(t - x\right) \cdot y}}{a - z} \]

      3. lower--.f64 81.0

         \[\leadsto x + \frac{\color{blue}{\left(t - x\right)} \cdot y}{a - z} \]

   5. Applied rewrites 81.0%

      \[\leadsto x + \frac{\color{blue}{\left(t - x\right) \cdot y}}{a - z} \]

   if 2.29999999999999989e-12 < z < 2.50000000000000013e31

   1. Initial program 69.6%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto x + \color{blue}{\frac{\left(t - x\right) \cdot \left(y - z\right)}{a}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto x + \color{blue}{\left(t - x\right) \cdot \frac{y - z}{a}} \]

      2. *-commutative N/A

         \[\leadsto x + \color{blue}{\frac{y - z}{a} \cdot \left(t - x\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto x + \color{blue}{\frac{y - z}{a} \cdot \left(t - x\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{y - z}{a}} \cdot \left(t - x\right) \]

      5. lower--.f64 N/A

         \[\leadsto x + \frac{\color{blue}{y - z}}{a} \cdot \left(t - x\right) \]

      6. lower--.f64 99.7

         \[\leadsto x + \frac{y - z}{a} \cdot \color{blue}{\left(t - x\right)} \]

   5. Applied rewrites 99.7%

      \[\leadsto x + \color{blue}{\frac{y - z}{a} \cdot \left(t - x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-\left(t - x\right)}{z}, y - a, t\right)\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-12}:\\ \;\;\;\;x + \frac{\left(t - x\right) \cdot y}{a - z}\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+31}:\\ \;\;\;\;x + \frac{y - z}{a} \cdot \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-\left(t - x\right)}{z}, y - a, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t - x, \frac{y - z}{a}, x\right)\\ \mathbf{if}\;a \leq -3.2 \cdot 10^{-22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 3.6 \cdot 10^{-132}:\\ \;\;\;\;t - \frac{\left(t - x\right) \cdot y}{z}\\ \mathbf{elif}\;a \leq 1.66 \cdot 10^{+102}:\\ \;\;\;\;t - \frac{-x}{z} \cdot \left(y - a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- t x) (/ (- y z) a) x)))
   (if (<= a -3.2e-22)
     t_1
     (if (<= a 3.6e-132)
       (- t (/ (* (- t x) y) z))
       (if (<= a 1.66e+102) (- t (* (/ (- x) z) (- y a))) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((t - x), ((y - z) / a), x);
	double tmp;
	if (a <= -3.2e-22) {
		tmp = t_1;
	} else if (a <= 3.6e-132) {
		tmp = t - (((t - x) * y) / z);
	} else if (a <= 1.66e+102) {
		tmp = t - ((-x / z) * (y - a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(t - x), Float64(Float64(y - z) / a), x)
	tmp = 0.0
	if (a <= -3.2e-22)
		tmp = t_1;
	elseif (a <= 3.6e-132)
		tmp = Float64(t - Float64(Float64(Float64(t - x) * y) / z));
	elseif (a <= 1.66e+102)
		tmp = Float64(t - Float64(Float64(Float64(-x) / z) * Float64(y - a)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t - x), $MachinePrecision] * N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -3.2e-22], t$95$1, If[LessEqual[a, 3.6e-132], N[(t - N[(N[(N[(t - x), $MachinePrecision] * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.66e+102], N[(t - N[(N[((-x) / z), $MachinePrecision] * N[(y - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.19999999999999987e-22 or 1.66e102 < a

   1. Initial program 69.8%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x + \frac{\left(t - x\right) \cdot \left(y - z\right)}{a}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(t - x\right) \cdot \left(y - z\right)}{a} + x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(y - z\right) \cdot \left(t - x\right)}}{a} + x \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a}} + x \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - z}, \frac{t - x}{a}, x\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y - z, \color{blue}{\frac{t - x}{a}}, x\right) \]

      7. lower--.f64 73.4

         \[\leadsto \mathsf{fma}\left(y - z, \frac{\color{blue}{t - x}}{a}, x\right) \]

   5. Applied rewrites 73.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 75.4%

      \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{\frac{y - z}{a}}, x\right) \]

   if -3.19999999999999987e-22 < a < 3.60000000000000007e-132

   1. Initial program 66.0%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(y - z\right) \cdot \left(t - x\right)}}{a - z} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t - x}{a - z} \cdot \left(y - z\right)} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

      8. lower-/.f64 69.3

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{t - x}{a - z}}, y - z, x\right) \]

   4. Applied rewrites 69.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{y \cdot \left(t - x\right)}{z}\right) - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}} \]
   6. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{y \cdot \left(t - x\right)}{z}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(t - x\right)}{z}} \]

      2. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(t - \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{y \cdot \left(t - x\right)}{z}\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(t - x\right)}{z} \]

      3. metadata-eval N/A

         \[\leadsto \left(t - \color{blue}{1} \cdot \frac{y \cdot \left(t - x\right)}{z}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(t - x\right)}{z} \]

      4. *-lft-identity N/A

         \[\leadsto \left(t - \color{blue}{\frac{y \cdot \left(t - x\right)}{z}}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(t - x\right)}{z} \]

      5. metadata-eval N/A

         \[\leadsto \left(t - \frac{y \cdot \left(t - x\right)}{z}\right) + \color{blue}{1} \cdot \frac{a \cdot \left(t - x\right)}{z} \]

      6. *-lft-identity N/A

         \[\leadsto \left(t - \frac{y \cdot \left(t - x\right)}{z}\right) + \color{blue}{\frac{a \cdot \left(t - x\right)}{z}} \]

      7. associate-+l- N/A

         \[\leadsto \color{blue}{t - \left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      8. div-sub N/A

         \[\leadsto t - \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      9. lower--.f64 N/A

         \[\leadsto \color{blue}{t - \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      10. div-sub N/A

          \[\leadsto t - \color{blue}{\left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      11. associate-/l* N/A

          \[\leadsto t - \left(\color{blue}{y \cdot \frac{t - x}{z}} - \frac{a \cdot \left(t - x\right)}{z}\right) \]

      12. associate-/l* N/A

          \[\leadsto t - \left(y \cdot \frac{t - x}{z} - \color{blue}{a \cdot \frac{t - x}{z}}\right) \]

      13. distribute-rgt-out-- N/A

          \[\leadsto t - \color{blue}{\frac{t - x}{z} \cdot \left(y - a\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto t - \color{blue}{\frac{t - x}{z} \cdot \left(y - a\right)} \]

      15. lower-/.f64 N/A

          \[\leadsto t - \color{blue}{\frac{t - x}{z}} \cdot \left(y - a\right) \]

      16. lower--.f64 N/A

          \[\leadsto t - \frac{\color{blue}{t - x}}{z} \cdot \left(y - a\right) \]

      17. lower--.f64 85.6

          \[\leadsto t - \frac{t - x}{z} \cdot \color{blue}{\left(y - a\right)} \]

   7. Applied rewrites 85.6%

      \[\leadsto \color{blue}{t - \frac{t - x}{z} \cdot \left(y - a\right)} \]

   8. Taylor expanded in y around inf

      \[\leadsto t - \frac{y \cdot \left(t - x\right)}{\color{blue}{z}} \]
   9. Step-by-step derivation

   10. Applied rewrites 83.8%

       \[\leadsto t - \frac{\left(t - x\right) \cdot y}{\color{blue}{z}} \]

   if 3.60000000000000007e-132 < a < 1.66e102

   1. Initial program 69.0%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(y - z\right) \cdot \left(t - x\right)}}{a - z} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t - x}{a - z} \cdot \left(y - z\right)} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

      8. lower-/.f64 74.5

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{t - x}{a - z}}, y - z, x\right) \]

   4. Applied rewrites 74.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{y \cdot \left(t - x\right)}{z}\right) - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}} \]
   6. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{y \cdot \left(t - x\right)}{z}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(t - x\right)}{z}} \]

      2. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(t - \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{y \cdot \left(t - x\right)}{z}\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(t - x\right)}{z} \]

      3. metadata-eval N/A

         \[\leadsto \left(t - \color{blue}{1} \cdot \frac{y \cdot \left(t - x\right)}{z}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(t - x\right)}{z} \]

      4. *-lft-identity N/A

         \[\leadsto \left(t - \color{blue}{\frac{y \cdot \left(t - x\right)}{z}}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(t - x\right)}{z} \]

      5. metadata-eval N/A

         \[\leadsto \left(t - \frac{y \cdot \left(t - x\right)}{z}\right) + \color{blue}{1} \cdot \frac{a \cdot \left(t - x\right)}{z} \]

      6. *-lft-identity N/A

         \[\leadsto \left(t - \frac{y \cdot \left(t - x\right)}{z}\right) + \color{blue}{\frac{a \cdot \left(t - x\right)}{z}} \]

      7. associate-+l- N/A

         \[\leadsto \color{blue}{t - \left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      8. div-sub N/A

         \[\leadsto t - \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      9. lower--.f64 N/A

         \[\leadsto \color{blue}{t - \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      10. div-sub N/A

          \[\leadsto t - \color{blue}{\left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      11. associate-/l* N/A

          \[\leadsto t - \left(\color{blue}{y \cdot \frac{t - x}{z}} - \frac{a \cdot \left(t - x\right)}{z}\right) \]

      12. associate-/l* N/A

          \[\leadsto t - \left(y \cdot \frac{t - x}{z} - \color{blue}{a \cdot \frac{t - x}{z}}\right) \]

      13. distribute-rgt-out-- N/A

          \[\leadsto t - \color{blue}{\frac{t - x}{z} \cdot \left(y - a\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto t - \color{blue}{\frac{t - x}{z} \cdot \left(y - a\right)} \]

      15. lower-/.f64 N/A

          \[\leadsto t - \color{blue}{\frac{t - x}{z}} \cdot \left(y - a\right) \]

      16. lower--.f64 N/A

          \[\leadsto t - \frac{\color{blue}{t - x}}{z} \cdot \left(y - a\right) \]

      17. lower--.f64 64.6

          \[\leadsto t - \frac{t - x}{z} \cdot \color{blue}{\left(y - a\right)} \]

   7. Applied rewrites 64.6%

      \[\leadsto \color{blue}{t - \frac{t - x}{z} \cdot \left(y - a\right)} \]

   8. Taylor expanded in x around inf

      \[\leadsto t - \left(-1 \cdot \frac{x}{z}\right) \cdot \left(\color{blue}{y} - a\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 62.8%

       \[\leadsto t - \frac{-x}{z} \cdot \left(\color{blue}{y} - a\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.2 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y - z}{a}, x\right)\\ \mathbf{elif}\;a \leq 3.6 \cdot 10^{-132}:\\ \;\;\;\;t - \frac{\left(t - x\right) \cdot y}{z}\\ \mathbf{elif}\;a \leq 1.66 \cdot 10^{+102}:\\ \;\;\;\;t - \frac{-x}{z} \cdot \left(y - a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y - z}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t - x, \frac{y - z}{a}, x\right)\\ \mathbf{if}\;a \leq -3.2 \cdot 10^{-22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 3.6 \cdot 10^{-132}:\\ \;\;\;\;t - \frac{\left(t - x\right) \cdot y}{z}\\ \mathbf{elif}\;a \leq 1.66 \cdot 10^{+102}:\\ \;\;\;\;t - \left(-x\right) \cdot \frac{y - a}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- t x) (/ (- y z) a) x)))
   (if (<= a -3.2e-22)
     t_1
     (if (<= a 3.6e-132)
       (- t (/ (* (- t x) y) z))
       (if (<= a 1.66e+102) (- t (* (- x) (/ (- y a) z))) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((t - x), ((y - z) / a), x);
	double tmp;
	if (a <= -3.2e-22) {
		tmp = t_1;
	} else if (a <= 3.6e-132) {
		tmp = t - (((t - x) * y) / z);
	} else if (a <= 1.66e+102) {
		tmp = t - (-x * ((y - a) / z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(t - x), Float64(Float64(y - z) / a), x)
	tmp = 0.0
	if (a <= -3.2e-22)
		tmp = t_1;
	elseif (a <= 3.6e-132)
		tmp = Float64(t - Float64(Float64(Float64(t - x) * y) / z));
	elseif (a <= 1.66e+102)
		tmp = Float64(t - Float64(Float64(-x) * Float64(Float64(y - a) / z)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t - x), $MachinePrecision] * N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -3.2e-22], t$95$1, If[LessEqual[a, 3.6e-132], N[(t - N[(N[(N[(t - x), $MachinePrecision] * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.66e+102], N[(t - N[((-x) * N[(N[(y - a), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.19999999999999987e-22 or 1.66e102 < a

   1. Initial program 69.8%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x + \frac{\left(t - x\right) \cdot \left(y - z\right)}{a}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(t - x\right) \cdot \left(y - z\right)}{a} + x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(y - z\right) \cdot \left(t - x\right)}}{a} + x \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a}} + x \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - z}, \frac{t - x}{a}, x\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y - z, \color{blue}{\frac{t - x}{a}}, x\right) \]

      7. lower--.f64 73.4

         \[\leadsto \mathsf{fma}\left(y - z, \frac{\color{blue}{t - x}}{a}, x\right) \]

   5. Applied rewrites 73.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 75.4%

      \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{\frac{y - z}{a}}, x\right) \]

   if -3.19999999999999987e-22 < a < 3.60000000000000007e-132

   1. Initial program 66.0%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(y - z\right) \cdot \left(t - x\right)}}{a - z} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t - x}{a - z} \cdot \left(y - z\right)} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

      8. lower-/.f64 69.3

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{t - x}{a - z}}, y - z, x\right) \]

   4. Applied rewrites 69.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{y \cdot \left(t - x\right)}{z}\right) - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}} \]
   6. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{y \cdot \left(t - x\right)}{z}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(t - x\right)}{z}} \]

      2. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(t - \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{y \cdot \left(t - x\right)}{z}\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(t - x\right)}{z} \]

      3. metadata-eval N/A

         \[\leadsto \left(t - \color{blue}{1} \cdot \frac{y \cdot \left(t - x\right)}{z}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(t - x\right)}{z} \]

      4. *-lft-identity N/A

         \[\leadsto \left(t - \color{blue}{\frac{y \cdot \left(t - x\right)}{z}}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(t - x\right)}{z} \]

      5. metadata-eval N/A

         \[\leadsto \left(t - \frac{y \cdot \left(t - x\right)}{z}\right) + \color{blue}{1} \cdot \frac{a \cdot \left(t - x\right)}{z} \]

      6. *-lft-identity N/A

         \[\leadsto \left(t - \frac{y \cdot \left(t - x\right)}{z}\right) + \color{blue}{\frac{a \cdot \left(t - x\right)}{z}} \]

      7. associate-+l- N/A

         \[\leadsto \color{blue}{t - \left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      8. div-sub N/A

         \[\leadsto t - \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      9. lower--.f64 N/A

         \[\leadsto \color{blue}{t - \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      10. div-sub N/A

          \[\leadsto t - \color{blue}{\left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      11. associate-/l* N/A

          \[\leadsto t - \left(\color{blue}{y \cdot \frac{t - x}{z}} - \frac{a \cdot \left(t - x\right)}{z}\right) \]

      12. associate-/l* N/A

          \[\leadsto t - \left(y \cdot \frac{t - x}{z} - \color{blue}{a \cdot \frac{t - x}{z}}\right) \]

      13. distribute-rgt-out-- N/A

          \[\leadsto t - \color{blue}{\frac{t - x}{z} \cdot \left(y - a\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto t - \color{blue}{\frac{t - x}{z} \cdot \left(y - a\right)} \]

      15. lower-/.f64 N/A

          \[\leadsto t - \color{blue}{\frac{t - x}{z}} \cdot \left(y - a\right) \]

      16. lower--.f64 N/A

          \[\leadsto t - \frac{\color{blue}{t - x}}{z} \cdot \left(y - a\right) \]

      17. lower--.f64 85.6

          \[\leadsto t - \frac{t - x}{z} \cdot \color{blue}{\left(y - a\right)} \]

   7. Applied rewrites 85.6%

      \[\leadsto \color{blue}{t - \frac{t - x}{z} \cdot \left(y - a\right)} \]

   8. Taylor expanded in y around inf

      \[\leadsto t - \frac{y \cdot \left(t - x\right)}{\color{blue}{z}} \]
   9. Step-by-step derivation

   10. Applied rewrites 83.8%

       \[\leadsto t - \frac{\left(t - x\right) \cdot y}{\color{blue}{z}} \]

   if 3.60000000000000007e-132 < a < 1.66e102

   1. Initial program 69.0%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(y - z\right) \cdot \left(t - x\right)}}{a - z} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t - x}{a - z} \cdot \left(y - z\right)} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

      8. lower-/.f64 74.5

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{t - x}{a - z}}, y - z, x\right) \]

   4. Applied rewrites 74.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{y \cdot \left(t - x\right)}{z}\right) - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}} \]
   6. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{y \cdot \left(t - x\right)}{z}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(t - x\right)}{z}} \]

      2. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(t - \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{y \cdot \left(t - x\right)}{z}\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(t - x\right)}{z} \]

      3. metadata-eval N/A

         \[\leadsto \left(t - \color{blue}{1} \cdot \frac{y \cdot \left(t - x\right)}{z}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(t - x\right)}{z} \]

      4. *-lft-identity N/A

         \[\leadsto \left(t - \color{blue}{\frac{y \cdot \left(t - x\right)}{z}}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(t - x\right)}{z} \]

      5. metadata-eval N/A

         \[\leadsto \left(t - \frac{y \cdot \left(t - x\right)}{z}\right) + \color{blue}{1} \cdot \frac{a \cdot \left(t - x\right)}{z} \]

      6. *-lft-identity N/A

         \[\leadsto \left(t - \frac{y \cdot \left(t - x\right)}{z}\right) + \color{blue}{\frac{a \cdot \left(t - x\right)}{z}} \]

      7. associate-+l- N/A

         \[\leadsto \color{blue}{t - \left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      8. div-sub N/A

         \[\leadsto t - \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      9. lower--.f64 N/A

         \[\leadsto \color{blue}{t - \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      10. div-sub N/A

          \[\leadsto t - \color{blue}{\left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      11. associate-/l* N/A

          \[\leadsto t - \left(\color{blue}{y \cdot \frac{t - x}{z}} - \frac{a \cdot \left(t - x\right)}{z}\right) \]

      12. associate-/l* N/A

          \[\leadsto t - \left(y \cdot \frac{t - x}{z} - \color{blue}{a \cdot \frac{t - x}{z}}\right) \]

      13. distribute-rgt-out-- N/A

          \[\leadsto t - \color{blue}{\frac{t - x}{z} \cdot \left(y - a\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto t - \color{blue}{\frac{t - x}{z} \cdot \left(y - a\right)} \]

      15. lower-/.f64 N/A

          \[\leadsto t - \color{blue}{\frac{t - x}{z}} \cdot \left(y - a\right) \]

      16. lower--.f64 N/A

          \[\leadsto t - \frac{\color{blue}{t - x}}{z} \cdot \left(y - a\right) \]

      17. lower--.f64 64.6

          \[\leadsto t - \frac{t - x}{z} \cdot \color{blue}{\left(y - a\right)} \]

   7. Applied rewrites 64.6%

      \[\leadsto \color{blue}{t - \frac{t - x}{z} \cdot \left(y - a\right)} \]

   8. Taylor expanded in x around inf

      \[\leadsto t - -1 \cdot \color{blue}{\frac{x \cdot \left(y - a\right)}{z}} \]
   9. Step-by-step derivation

   10. Applied rewrites 62.8%

       \[\leadsto t - \left(-x\right) \cdot \color{blue}{\frac{y - a}{z}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.2 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y - z}{a}, x\right)\\ \mathbf{elif}\;a \leq 3.6 \cdot 10^{-132}:\\ \;\;\;\;t - \frac{\left(t - x\right) \cdot y}{z}\\ \mathbf{elif}\;a \leq 1.66 \cdot 10^{+102}:\\ \;\;\;\;t - \left(-x\right) \cdot \frac{y - a}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y - z}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t - x, \frac{y - z}{a}, x\right)\\ \mathbf{if}\;a \leq -3.2 \cdot 10^{-22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 2.35 \cdot 10^{-153}:\\ \;\;\;\;t - \frac{\left(t - x\right) \cdot y}{z}\\ \mathbf{elif}\;a \leq 1.75 \cdot 10^{+102}:\\ \;\;\;\;\frac{y - z}{a - z} \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- t x) (/ (- y z) a) x)))
   (if (<= a -3.2e-22)
     t_1
     (if (<= a 2.35e-153)
       (- t (/ (* (- t x) y) z))
       (if (<= a 1.75e+102) (* (/ (- y z) (- a z)) t) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((t - x), ((y - z) / a), x);
	double tmp;
	if (a <= -3.2e-22) {
		tmp = t_1;
	} else if (a <= 2.35e-153) {
		tmp = t - (((t - x) * y) / z);
	} else if (a <= 1.75e+102) {
		tmp = ((y - z) / (a - z)) * t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(t - x), Float64(Float64(y - z) / a), x)
	tmp = 0.0
	if (a <= -3.2e-22)
		tmp = t_1;
	elseif (a <= 2.35e-153)
		tmp = Float64(t - Float64(Float64(Float64(t - x) * y) / z));
	elseif (a <= 1.75e+102)
		tmp = Float64(Float64(Float64(y - z) / Float64(a - z)) * t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t - x), $MachinePrecision] * N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -3.2e-22], t$95$1, If[LessEqual[a, 2.35e-153], N[(t - N[(N[(N[(t - x), $MachinePrecision] * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.75e+102], N[(N[(N[(y - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.19999999999999987e-22 or 1.75000000000000005e102 < a

   1. Initial program 69.8%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x + \frac{\left(t - x\right) \cdot \left(y - z\right)}{a}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(t - x\right) \cdot \left(y - z\right)}{a} + x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(y - z\right) \cdot \left(t - x\right)}}{a} + x \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a}} + x \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - z}, \frac{t - x}{a}, x\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y - z, \color{blue}{\frac{t - x}{a}}, x\right) \]

      7. lower--.f64 73.4

         \[\leadsto \mathsf{fma}\left(y - z, \frac{\color{blue}{t - x}}{a}, x\right) \]

   5. Applied rewrites 73.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 75.4%

      \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{\frac{y - z}{a}}, x\right) \]

   if -3.19999999999999987e-22 < a < 2.35e-153

   1. Initial program 64.8%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(y - z\right) \cdot \left(t - x\right)}}{a - z} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t - x}{a - z} \cdot \left(y - z\right)} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

      8. lower-/.f64 68.2

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{t - x}{a - z}}, y - z, x\right) \]

   4. Applied rewrites 68.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{y \cdot \left(t - x\right)}{z}\right) - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}} \]
   6. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{y \cdot \left(t - x\right)}{z}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(t - x\right)}{z}} \]

      2. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(t - \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{y \cdot \left(t - x\right)}{z}\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(t - x\right)}{z} \]

      3. metadata-eval N/A

         \[\leadsto \left(t - \color{blue}{1} \cdot \frac{y \cdot \left(t - x\right)}{z}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(t - x\right)}{z} \]

      4. *-lft-identity N/A

         \[\leadsto \left(t - \color{blue}{\frac{y \cdot \left(t - x\right)}{z}}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(t - x\right)}{z} \]

      5. metadata-eval N/A

         \[\leadsto \left(t - \frac{y \cdot \left(t - x\right)}{z}\right) + \color{blue}{1} \cdot \frac{a \cdot \left(t - x\right)}{z} \]

      6. *-lft-identity N/A

         \[\leadsto \left(t - \frac{y \cdot \left(t - x\right)}{z}\right) + \color{blue}{\frac{a \cdot \left(t - x\right)}{z}} \]

      7. associate-+l- N/A

         \[\leadsto \color{blue}{t - \left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      8. div-sub N/A

         \[\leadsto t - \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      9. lower--.f64 N/A

         \[\leadsto \color{blue}{t - \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      10. div-sub N/A

          \[\leadsto t - \color{blue}{\left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      11. associate-/l* N/A

          \[\leadsto t - \left(\color{blue}{y \cdot \frac{t - x}{z}} - \frac{a \cdot \left(t - x\right)}{z}\right) \]

      12. associate-/l* N/A

          \[\leadsto t - \left(y \cdot \frac{t - x}{z} - \color{blue}{a \cdot \frac{t - x}{z}}\right) \]

      13. distribute-rgt-out-- N/A

          \[\leadsto t - \color{blue}{\frac{t - x}{z} \cdot \left(y - a\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto t - \color{blue}{\frac{t - x}{z} \cdot \left(y - a\right)} \]

      15. lower-/.f64 N/A

          \[\leadsto t - \color{blue}{\frac{t - x}{z}} \cdot \left(y - a\right) \]

      16. lower--.f64 N/A

          \[\leadsto t - \frac{\color{blue}{t - x}}{z} \cdot \left(y - a\right) \]

      17. lower--.f64 86.0

          \[\leadsto t - \frac{t - x}{z} \cdot \color{blue}{\left(y - a\right)} \]

   7. Applied rewrites 86.0%

      \[\leadsto \color{blue}{t - \frac{t - x}{z} \cdot \left(y - a\right)} \]

   8. Taylor expanded in y around inf

      \[\leadsto t - \frac{y \cdot \left(t - x\right)}{\color{blue}{z}} \]
   9. Step-by-step derivation

   10. Applied rewrites 84.0%

       \[\leadsto t - \frac{\left(t - x\right) \cdot y}{\color{blue}{z}} \]

   if 2.35e-153 < a < 1.75000000000000005e102

   1. Initial program 71.2%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(y - z\right) \cdot \left(t - x\right)}}{a - z} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t - x}{a - z} \cdot \left(y - z\right)} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

      8. lower-/.f64 76.3

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{t - x}{a - z}}, y - z, x\right) \]

   4. Applied rewrites 76.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(\left(-1 \cdot \frac{x \cdot \left(y - z\right)}{t \cdot \left(a - z\right)} + \left(\frac{x}{t} + \frac{y}{a - z}\right)\right) - \frac{z}{a - z}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot \frac{x \cdot \left(y - z\right)}{t \cdot \left(a - z\right)} + \left(\frac{x}{t} + \frac{y}{a - z}\right)\right) - \frac{z}{a - z}\right) \cdot t} \]

   7. Applied rewrites 82.1%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left(-x, \frac{y - z}{a - z}, x\right)}{t} + \frac{y - z}{a - z}\right) \cdot t} \]

   8. Taylor expanded in x around 0

      \[\leadsto \left(\frac{y}{a - z} - \frac{z}{a - z}\right) \cdot t \]
   9. Step-by-step derivation

   10. Applied rewrites 63.3%

       \[\leadsto \frac{y - z}{a - z} \cdot t \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.2 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y - z}{a}, x\right)\\ \mathbf{elif}\;a \leq 2.35 \cdot 10^{-153}:\\ \;\;\;\;t - \frac{\left(t - x\right) \cdot y}{z}\\ \mathbf{elif}\;a \leq 1.75 \cdot 10^{+102}:\\ \;\;\;\;\frac{y - z}{a - z} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y - z}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y - z, \frac{t}{a}, x\right)\\ \mathbf{if}\;a \leq -1.28 \cdot 10^{+135}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 2.35 \cdot 10^{-153}:\\ \;\;\;\;t - \frac{\left(t - x\right) \cdot y}{z}\\ \mathbf{elif}\;a \leq 4.1 \cdot 10^{+102}:\\ \;\;\;\;\frac{y - z}{a - z} \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- y z) (/ t a) x)))
   (if (<= a -1.28e+135)
     t_1
     (if (<= a 2.35e-153)
       (- t (/ (* (- t x) y) z))
       (if (<= a 4.1e+102) (* (/ (- y z) (- a z)) t) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((y - z), (t / a), x);
	double tmp;
	if (a <= -1.28e+135) {
		tmp = t_1;
	} else if (a <= 2.35e-153) {
		tmp = t - (((t - x) * y) / z);
	} else if (a <= 4.1e+102) {
		tmp = ((y - z) / (a - z)) * t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(y - z), Float64(t / a), x)
	tmp = 0.0
	if (a <= -1.28e+135)
		tmp = t_1;
	elseif (a <= 2.35e-153)
		tmp = Float64(t - Float64(Float64(Float64(t - x) * y) / z));
	elseif (a <= 4.1e+102)
		tmp = Float64(Float64(Float64(y - z) / Float64(a - z)) * t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - z), $MachinePrecision] * N[(t / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -1.28e+135], t$95$1, If[LessEqual[a, 2.35e-153], N[(t - N[(N[(N[(t - x), $MachinePrecision] * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 4.1e+102], N[(N[(N[(y - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.28e135 or 4.1e102 < a

   1. Initial program 73.9%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x + \frac{\left(t - x\right) \cdot \left(y - z\right)}{a}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(t - x\right) \cdot \left(y - z\right)}{a} + x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(y - z\right) \cdot \left(t - x\right)}}{a} + x \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a}} + x \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - z}, \frac{t - x}{a}, x\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y - z, \color{blue}{\frac{t - x}{a}}, x\right) \]

      7. lower--.f64 82.1

         \[\leadsto \mathsf{fma}\left(y - z, \frac{\color{blue}{t - x}}{a}, x\right) \]

   5. Applied rewrites 82.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(y - z, \frac{t}{\color{blue}{a}}, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 79.2%

      \[\leadsto \mathsf{fma}\left(y - z, \frac{t}{\color{blue}{a}}, x\right) \]

   if -1.28e135 < a < 2.35e-153

   1. Initial program 63.7%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(y - z\right) \cdot \left(t - x\right)}}{a - z} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t - x}{a - z} \cdot \left(y - z\right)} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

      8. lower-/.f64 70.0

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{t - x}{a - z}}, y - z, x\right) \]

   4. Applied rewrites 70.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{y \cdot \left(t - x\right)}{z}\right) - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}} \]
   6. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{y \cdot \left(t - x\right)}{z}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(t - x\right)}{z}} \]

      2. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(t - \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{y \cdot \left(t - x\right)}{z}\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(t - x\right)}{z} \]

      3. metadata-eval N/A

         \[\leadsto \left(t - \color{blue}{1} \cdot \frac{y \cdot \left(t - x\right)}{z}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(t - x\right)}{z} \]

      4. *-lft-identity N/A

         \[\leadsto \left(t - \color{blue}{\frac{y \cdot \left(t - x\right)}{z}}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(t - x\right)}{z} \]

      5. metadata-eval N/A

         \[\leadsto \left(t - \frac{y \cdot \left(t - x\right)}{z}\right) + \color{blue}{1} \cdot \frac{a \cdot \left(t - x\right)}{z} \]

      6. *-lft-identity N/A

         \[\leadsto \left(t - \frac{y \cdot \left(t - x\right)}{z}\right) + \color{blue}{\frac{a \cdot \left(t - x\right)}{z}} \]

      7. associate-+l- N/A

         \[\leadsto \color{blue}{t - \left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      8. div-sub N/A

         \[\leadsto t - \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      9. lower--.f64 N/A

         \[\leadsto \color{blue}{t - \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      10. div-sub N/A

          \[\leadsto t - \color{blue}{\left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      11. associate-/l* N/A

          \[\leadsto t - \left(\color{blue}{y \cdot \frac{t - x}{z}} - \frac{a \cdot \left(t - x\right)}{z}\right) \]

      12. associate-/l* N/A

          \[\leadsto t - \left(y \cdot \frac{t - x}{z} - \color{blue}{a \cdot \frac{t - x}{z}}\right) \]

      13. distribute-rgt-out-- N/A

          \[\leadsto t - \color{blue}{\frac{t - x}{z} \cdot \left(y - a\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto t - \color{blue}{\frac{t - x}{z} \cdot \left(y - a\right)} \]

      15. lower-/.f64 N/A

          \[\leadsto t - \color{blue}{\frac{t - x}{z}} \cdot \left(y - a\right) \]

      16. lower--.f64 N/A

          \[\leadsto t - \frac{\color{blue}{t - x}}{z} \cdot \left(y - a\right) \]

      17. lower--.f64 81.6

          \[\leadsto t - \frac{t - x}{z} \cdot \color{blue}{\left(y - a\right)} \]

   7. Applied rewrites 81.6%

      \[\leadsto \color{blue}{t - \frac{t - x}{z} \cdot \left(y - a\right)} \]

   8. Taylor expanded in y around inf

      \[\leadsto t - \frac{y \cdot \left(t - x\right)}{\color{blue}{z}} \]
   9. Step-by-step derivation

   10. Applied rewrites 76.4%

       \[\leadsto t - \frac{\left(t - x\right) \cdot y}{\color{blue}{z}} \]

   if 2.35e-153 < a < 4.1e102

   1. Initial program 71.2%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(y - z\right) \cdot \left(t - x\right)}}{a - z} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t - x}{a - z} \cdot \left(y - z\right)} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

      8. lower-/.f64 76.3

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{t - x}{a - z}}, y - z, x\right) \]

   4. Applied rewrites 76.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(\left(-1 \cdot \frac{x \cdot \left(y - z\right)}{t \cdot \left(a - z\right)} + \left(\frac{x}{t} + \frac{y}{a - z}\right)\right) - \frac{z}{a - z}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot \frac{x \cdot \left(y - z\right)}{t \cdot \left(a - z\right)} + \left(\frac{x}{t} + \frac{y}{a - z}\right)\right) - \frac{z}{a - z}\right) \cdot t} \]

   7. Applied rewrites 82.1%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left(-x, \frac{y - z}{a - z}, x\right)}{t} + \frac{y - z}{a - z}\right) \cdot t} \]

   8. Taylor expanded in x around 0

      \[\leadsto \left(\frac{y}{a - z} - \frac{z}{a - z}\right) \cdot t \]
   9. Step-by-step derivation

   10. Applied rewrites 63.3%

       \[\leadsto \frac{y - z}{a - z} \cdot t \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.28 \cdot 10^{+135}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t}{a}, x\right)\\ \mathbf{elif}\;a \leq 2.35 \cdot 10^{-153}:\\ \;\;\;\;t - \frac{\left(t - x\right) \cdot y}{z}\\ \mathbf{elif}\;a \leq 4.1 \cdot 10^{+102}:\\ \;\;\;\;\frac{y - z}{a - z} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 58.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y - z\right) \cdot \frac{t}{a - z}\\ \mathbf{if}\;t \leq -9.8 \cdot 10^{-47}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -1.45 \cdot 10^{-262}:\\ \;\;\;\;\mathsf{fma}\left(-x, \frac{y}{a}, x\right)\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-40}:\\ \;\;\;\;\left(t - x\right) \cdot \frac{y}{a - z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (- y z) (/ t (- a z)))))
   (if (<= t -9.8e-47)
     t_1
     (if (<= t -1.45e-262)
       (fma (- x) (/ y a) x)
       (if (<= t 7.5e-40) (* (- t x) (/ y (- a z))) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y - z) * (t / (a - z));
	double tmp;
	if (t <= -9.8e-47) {
		tmp = t_1;
	} else if (t <= -1.45e-262) {
		tmp = fma(-x, (y / a), x);
	} else if (t <= 7.5e-40) {
		tmp = (t - x) * (y / (a - z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y - z) * Float64(t / Float64(a - z)))
	tmp = 0.0
	if (t <= -9.8e-47)
		tmp = t_1;
	elseif (t <= -1.45e-262)
		tmp = fma(Float64(-x), Float64(y / a), x);
	elseif (t <= 7.5e-40)
		tmp = Float64(Float64(t - x) * Float64(y / Float64(a - z)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - z), $MachinePrecision] * N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -9.8e-47], t$95$1, If[LessEqual[t, -1.45e-262], N[((-x) * N[(y / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t, 7.5e-40], N[(N[(t - x), $MachinePrecision] * N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -9.800000000000001e-47 or 7.50000000000000069e-40 < t

   1. Initial program 68.2%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(y - z\right) \cdot t}}{a - z} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t}{a - z}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t}{a - z}} \]

      4. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(y - z\right)} \cdot \frac{t}{a - z} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{t}{a - z}} \]

      6. lower--.f64 72.9

         \[\leadsto \left(y - z\right) \cdot \frac{t}{\color{blue}{a - z}} \]

   5. Applied rewrites 72.9%

      \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t}{a - z}} \]

   if -9.800000000000001e-47 < t < -1.44999999999999998e-262

   1. Initial program 71.9%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(y - z\right) \cdot \left(t - x\right)}}{a - z} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t - x}{a - z} \cdot \left(y - z\right)} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

      8. lower-/.f64 64.1

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{t - x}{a - z}}, y - z, x\right) \]

   4. Applied rewrites 64.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + -1 \cdot \frac{y - z}{a - z}\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \frac{y - z}{a - z} + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \frac{y - z}{a - z}\right) + x \cdot 1} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{y - z}{a - z}\right)\right)} + x \cdot 1 \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{y - z}{a - z}\right)\right)} + x \cdot 1 \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{y - z}{a - z}} + x \cdot 1 \]

      6. *-rgt-identity N/A

         \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \frac{y - z}{a - z} + \color{blue}{x} \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{y - z}{a - z}, x\right)} \]

      8. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-x}, \frac{y - z}{a - z}, x\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(-x, \color{blue}{\frac{y - z}{a - z}}, x\right) \]

      10. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(-x, \frac{\color{blue}{y - z}}{a - z}, x\right) \]

      11. lower--.f64 58.2

          \[\leadsto \mathsf{fma}\left(-x, \frac{y - z}{\color{blue}{a - z}}, x\right) \]

   7. Applied rewrites 58.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-x, \frac{y - z}{a - z}, x\right)} \]

   8. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(-x, \frac{y}{\color{blue}{a}}, x\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 51.9%

       \[\leadsto \mathsf{fma}\left(-x, \frac{y}{\color{blue}{a}}, x\right) \]

   if -1.44999999999999998e-262 < t < 7.50000000000000069e-40

   1. Initial program 64.8%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{t}{a - z} - \frac{x}{a - z}\right)} \]
   4. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto y \cdot \color{blue}{\frac{t - x}{a - z}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(t - x\right)}{a - z}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(t - x\right) \cdot y}}{a - z} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\left(t - x\right) \cdot \frac{y}{a - z}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(t - x\right) \cdot \frac{y}{a - z}} \]

      6. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(t - x\right)} \cdot \frac{y}{a - z} \]

      7. lower-/.f64 N/A

         \[\leadsto \left(t - x\right) \cdot \color{blue}{\frac{y}{a - z}} \]

      8. lower--.f64 53.0

         \[\leadsto \left(t - x\right) \cdot \frac{y}{\color{blue}{a - z}} \]

   5. Applied rewrites 53.0%

      \[\leadsto \color{blue}{\left(t - x\right) \cdot \frac{y}{a - z}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9.8 \cdot 10^{-47}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{t}{a - z}\\ \mathbf{elif}\;t \leq -1.45 \cdot 10^{-262}:\\ \;\;\;\;\mathsf{fma}\left(-x, \frac{y}{a}, x\right)\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-40}:\\ \;\;\;\;\left(t - x\right) \cdot \frac{y}{a - z}\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{t}{a - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 60.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.8 \cdot 10^{+48}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t}{a}, x\right)\\ \mathbf{elif}\;a \leq -5.3 \cdot 10^{-197}:\\ \;\;\;\;\left(t - x\right) \cdot \frac{y}{a - z}\\ \mathbf{elif}\;a \leq 1.75 \cdot 10^{-9}:\\ \;\;\;\;\left(-t\right) \cdot \frac{y - z}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -4.8e+48)
   (fma (- y z) (/ t a) x)
   (if (<= a -5.3e-197)
     (* (- t x) (/ y (- a z)))
     (if (<= a 1.75e-9) (* (- t) (/ (- y z) z)) (fma (/ (- t x) a) y x)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -4.8e+48) {
		tmp = fma((y - z), (t / a), x);
	} else if (a <= -5.3e-197) {
		tmp = (t - x) * (y / (a - z));
	} else if (a <= 1.75e-9) {
		tmp = -t * ((y - z) / z);
	} else {
		tmp = fma(((t - x) / a), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -4.8e+48)
		tmp = fma(Float64(y - z), Float64(t / a), x);
	elseif (a <= -5.3e-197)
		tmp = Float64(Float64(t - x) * Float64(y / Float64(a - z)));
	elseif (a <= 1.75e-9)
		tmp = Float64(Float64(-t) * Float64(Float64(y - z) / z));
	else
		tmp = fma(Float64(Float64(t - x) / a), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -4.8e+48], N[(N[(y - z), $MachinePrecision] * N[(t / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[a, -5.3e-197], N[(N[(t - x), $MachinePrecision] * N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.75e-9], N[((-t) * N[(N[(y - z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] * y + x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if a < -4.8000000000000002e48

   1. Initial program 73.2%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x + \frac{\left(t - x\right) \cdot \left(y - z\right)}{a}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(t - x\right) \cdot \left(y - z\right)}{a} + x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(y - z\right) \cdot \left(t - x\right)}}{a} + x \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a}} + x \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - z}, \frac{t - x}{a}, x\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y - z, \color{blue}{\frac{t - x}{a}}, x\right) \]

      7. lower--.f64 70.6

         \[\leadsto \mathsf{fma}\left(y - z, \frac{\color{blue}{t - x}}{a}, x\right) \]

   5. Applied rewrites 70.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(y - z, \frac{t}{\color{blue}{a}}, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 68.6%

      \[\leadsto \mathsf{fma}\left(y - z, \frac{t}{\color{blue}{a}}, x\right) \]

   if -4.8000000000000002e48 < a < -5.29999999999999972e-197

   1. Initial program 67.4%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{t}{a - z} - \frac{x}{a - z}\right)} \]
   4. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto y \cdot \color{blue}{\frac{t - x}{a - z}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(t - x\right)}{a - z}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(t - x\right) \cdot y}}{a - z} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\left(t - x\right) \cdot \frac{y}{a - z}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(t - x\right) \cdot \frac{y}{a - z}} \]

      6. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(t - x\right)} \cdot \frac{y}{a - z} \]

      7. lower-/.f64 N/A

         \[\leadsto \left(t - x\right) \cdot \color{blue}{\frac{y}{a - z}} \]

      8. lower--.f64 60.7

         \[\leadsto \left(t - x\right) \cdot \frac{y}{\color{blue}{a - z}} \]

   5. Applied rewrites 60.7%

      \[\leadsto \color{blue}{\left(t - x\right) \cdot \frac{y}{a - z}} \]

   if -5.29999999999999972e-197 < a < 1.75e-9

   1. Initial program 66.6%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(y - z\right) \cdot \left(t - x\right)}}{a - z} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t - x}{a - z} \cdot \left(y - z\right)} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

      8. lower-/.f64 68.0

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{t - x}{a - z}}, y - z, x\right) \]

   4. Applied rewrites 68.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + \left(-1 \cdot \frac{y - z}{a - z} + \frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + \left(-1 \cdot \frac{y - z}{a - z} + \frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)}\right)\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \left(-1 \cdot \frac{y - z}{a - z} + \frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)}\right)\right) \cdot x} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot \frac{y - z}{a - z} + \frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)}\right) + 1\right)} \cdot x \]

      4. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)} + -1 \cdot \frac{y - z}{a - z}\right)} + 1\right) \cdot x \]

      5. times-frac N/A

         \[\leadsto \left(\left(\color{blue}{\frac{t}{x} \cdot \frac{y - z}{a - z}} + -1 \cdot \frac{y - z}{a - z}\right) + 1\right) \cdot x \]

      6. distribute-rgt-out N/A

         \[\leadsto \left(\color{blue}{\frac{y - z}{a - z} \cdot \left(\frac{t}{x} + -1\right)} + 1\right) \cdot x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, \frac{t}{x} + -1, 1\right)} \cdot x \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, \frac{t}{x} + -1, 1\right) \cdot x \]

      9. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{y - z}}{a - z}, \frac{t}{x} + -1, 1\right) \cdot x \]

      10. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{\color{blue}{a - z}}, \frac{t}{x} + -1, 1\right) \cdot x \]

      11. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{a - z}, \color{blue}{\frac{t}{x} + -1}, 1\right) \cdot x \]

      12. lower-/.f64 63.6

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{a - z}, \color{blue}{\frac{t}{x}} + -1, 1\right) \cdot x \]

   7. Applied rewrites 63.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, \frac{t}{x} + -1, 1\right) \cdot x} \]

   8. Taylor expanded in t around inf

      \[\leadsto t \cdot \color{blue}{\left(x \cdot \left(\frac{y}{x \cdot \left(a - z\right)} - \frac{z}{x \cdot \left(a - z\right)}\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 41.2%

       \[\leadsto \left(t \cdot x\right) \cdot \color{blue}{\frac{\frac{y - z}{x}}{a - z}} \]

   11. Taylor expanded in a around 0

       \[\leadsto -1 \cdot \frac{t \cdot \left(y - z\right)}{\color{blue}{z}} \]
   12. Step-by-step derivation

   13. Applied rewrites 60.8%

       \[\leadsto \left(-t\right) \cdot \frac{y - z}{\color{blue}{z}} \]

   if 1.75e-9 < a

   1. Initial program 66.8%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{y \cdot \left(t - x\right)}{a}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(t - x\right)}{a} + x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{t - x}{a}} + x \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t - x}{a} \cdot y} + x \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{t - x}{a}}, y, x\right) \]

      6. lower--.f64 63.6

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{t - x}}{a}, y, x\right) \]

   5. Applied rewrites 63.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.8 \cdot 10^{+48}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t}{a}, x\right)\\ \mathbf{elif}\;a \leq -5.3 \cdot 10^{-197}:\\ \;\;\;\;\left(t - x\right) \cdot \frac{y}{a - z}\\ \mathbf{elif}\;a \leq 1.75 \cdot 10^{-9}:\\ \;\;\;\;\left(-t\right) \cdot \frac{y - z}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 72.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y - z}{a}\\ \mathbf{if}\;a \leq -1.28 \cdot 10^{+135}:\\ \;\;\;\;x + t\_1 \cdot \left(t - x\right)\\ \mathbf{elif}\;a \leq 1.66 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-\left(t - x\right)}{z}, y - a, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, t\_1, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- y z) a)))
   (if (<= a -1.28e+135)
     (+ x (* t_1 (- t x)))
     (if (<= a 1.66e+102)
       (fma (/ (- (- t x)) z) (- y a) t)
       (fma (- t x) t_1 x)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y - z) / a;
	double tmp;
	if (a <= -1.28e+135) {
		tmp = x + (t_1 * (t - x));
	} else if (a <= 1.66e+102) {
		tmp = fma((-(t - x) / z), (y - a), t);
	} else {
		tmp = fma((t - x), t_1, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y - z) / a)
	tmp = 0.0
	if (a <= -1.28e+135)
		tmp = Float64(x + Float64(t_1 * Float64(t - x)));
	elseif (a <= 1.66e+102)
		tmp = fma(Float64(Float64(-Float64(t - x)) / z), Float64(y - a), t);
	else
		tmp = fma(Float64(t - x), t_1, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[a, -1.28e+135], N[(x + N[(t$95$1 * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.66e+102], N[(N[((-N[(t - x), $MachinePrecision]) / z), $MachinePrecision] * N[(y - a), $MachinePrecision] + t), $MachinePrecision], N[(N[(t - x), $MachinePrecision] * t$95$1 + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.28e135

   1. Initial program 78.7%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto x + \color{blue}{\frac{\left(t - x\right) \cdot \left(y - z\right)}{a}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto x + \color{blue}{\left(t - x\right) \cdot \frac{y - z}{a}} \]

      2. *-commutative N/A

         \[\leadsto x + \color{blue}{\frac{y - z}{a} \cdot \left(t - x\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto x + \color{blue}{\frac{y - z}{a} \cdot \left(t - x\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{y - z}{a}} \cdot \left(t - x\right) \]

      5. lower--.f64 N/A

         \[\leadsto x + \frac{\color{blue}{y - z}}{a} \cdot \left(t - x\right) \]

      6. lower--.f64 84.0

         \[\leadsto x + \frac{y - z}{a} \cdot \color{blue}{\left(t - x\right)} \]

   5. Applied rewrites 84.0%

      \[\leadsto x + \color{blue}{\frac{y - z}{a} \cdot \left(t - x\right)} \]

   if -1.28e135 < a < 1.66e102

   1. Initial program 65.9%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{y \cdot \left(t - x\right)}{z}\right) - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{t + \left(-1 \cdot \frac{y \cdot \left(t - x\right)}{z} - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      2. distribute-lft-out-- N/A

         \[\leadsto t + \color{blue}{-1 \cdot \left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      3. div-sub N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + t} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right)} + t \]

      6. div-sub N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)}\right)\right) + t \]

      7. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\left(\color{blue}{y \cdot \frac{t - x}{z}} - \frac{a \cdot \left(t - x\right)}{z}\right)\right)\right) + t \]

      8. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\left(y \cdot \frac{t - x}{z} - \color{blue}{a \cdot \frac{t - x}{z}}\right)\right)\right) + t \]

      9. distribute-rgt-out-- N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{t - x}{z} \cdot \left(y - a\right)}\right)\right) + t \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{t - x}{z}\right)\right) \cdot \left(y - a\right)} + t \]

      11. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\frac{t - x}{z}\right), y - a, t\right)} \]

   5. Applied rewrites 76.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-\left(t - x\right)}{z}, y - a, t\right)} \]

   if 1.66e102 < a

   1. Initial program 69.4%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x + \frac{\left(t - x\right) \cdot \left(y - z\right)}{a}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(t - x\right) \cdot \left(y - z\right)}{a} + x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(y - z\right) \cdot \left(t - x\right)}}{a} + x \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a}} + x \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - z}, \frac{t - x}{a}, x\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y - z, \color{blue}{\frac{t - x}{a}}, x\right) \]

      7. lower--.f64 80.3

         \[\leadsto \mathsf{fma}\left(y - z, \frac{\color{blue}{t - x}}{a}, x\right) \]

   5. Applied rewrites 80.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 85.9%

      \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{\frac{y - z}{a}}, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.28 \cdot 10^{+135}:\\ \;\;\;\;x + \frac{y - z}{a} \cdot \left(t - x\right)\\ \mathbf{elif}\;a \leq 1.66 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-\left(t - x\right)}{z}, y - a, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y - z}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 72.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.28 \cdot 10^{+135}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)\\ \mathbf{elif}\;a \leq 1.66 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-\left(t - x\right)}{z}, y - a, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y - z}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -1.28e+135)
   (fma (- y z) (/ (- t x) a) x)
   (if (<= a 1.66e+102)
     (fma (/ (- (- t x)) z) (- y a) t)
     (fma (- t x) (/ (- y z) a) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -1.28e+135) {
		tmp = fma((y - z), ((t - x) / a), x);
	} else if (a <= 1.66e+102) {
		tmp = fma((-(t - x) / z), (y - a), t);
	} else {
		tmp = fma((t - x), ((y - z) / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -1.28e+135)
		tmp = fma(Float64(y - z), Float64(Float64(t - x) / a), x);
	elseif (a <= 1.66e+102)
		tmp = fma(Float64(Float64(-Float64(t - x)) / z), Float64(y - a), t);
	else
		tmp = fma(Float64(t - x), Float64(Float64(y - z) / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -1.28e+135], N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[a, 1.66e+102], N[(N[((-N[(t - x), $MachinePrecision]) / z), $MachinePrecision] * N[(y - a), $MachinePrecision] + t), $MachinePrecision], N[(N[(t - x), $MachinePrecision] * N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.28e135

   1. Initial program 78.7%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x + \frac{\left(t - x\right) \cdot \left(y - z\right)}{a}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(t - x\right) \cdot \left(y - z\right)}{a} + x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(y - z\right) \cdot \left(t - x\right)}}{a} + x \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a}} + x \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - z}, \frac{t - x}{a}, x\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y - z, \color{blue}{\frac{t - x}{a}}, x\right) \]

      7. lower--.f64 83.9

         \[\leadsto \mathsf{fma}\left(y - z, \frac{\color{blue}{t - x}}{a}, x\right) \]

   5. Applied rewrites 83.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)} \]

   if -1.28e135 < a < 1.66e102

   1. Initial program 65.9%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{y \cdot \left(t - x\right)}{z}\right) - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{t + \left(-1 \cdot \frac{y \cdot \left(t - x\right)}{z} - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      2. distribute-lft-out-- N/A

         \[\leadsto t + \color{blue}{-1 \cdot \left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      3. div-sub N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + t} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right)} + t \]

      6. div-sub N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)}\right)\right) + t \]

      7. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\left(\color{blue}{y \cdot \frac{t - x}{z}} - \frac{a \cdot \left(t - x\right)}{z}\right)\right)\right) + t \]

      8. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\left(y \cdot \frac{t - x}{z} - \color{blue}{a \cdot \frac{t - x}{z}}\right)\right)\right) + t \]

      9. distribute-rgt-out-- N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{t - x}{z} \cdot \left(y - a\right)}\right)\right) + t \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{t - x}{z}\right)\right) \cdot \left(y - a\right)} + t \]

      11. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\frac{t - x}{z}\right), y - a, t\right)} \]

   5. Applied rewrites 76.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-\left(t - x\right)}{z}, y - a, t\right)} \]

   if 1.66e102 < a

   1. Initial program 69.4%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x + \frac{\left(t - x\right) \cdot \left(y - z\right)}{a}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(t - x\right) \cdot \left(y - z\right)}{a} + x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(y - z\right) \cdot \left(t - x\right)}}{a} + x \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a}} + x \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - z}, \frac{t - x}{a}, x\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y - z, \color{blue}{\frac{t - x}{a}}, x\right) \]

      7. lower--.f64 80.3

         \[\leadsto \mathsf{fma}\left(y - z, \frac{\color{blue}{t - x}}{a}, x\right) \]

   5. Applied rewrites 80.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 85.9%

      \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{\frac{y - z}{a}}, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.28 \cdot 10^{+135}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)\\ \mathbf{elif}\;a \leq 1.66 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-\left(t - x\right)}{z}, y - a, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y - z}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 63.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.05 \cdot 10^{+156} \lor \neg \left(a \leq 4.1 \cdot 10^{+102}\right):\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z}{a - z} \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -1.05e+156) (not (<= a 4.1e+102)))
   (fma (- y z) (/ t a) x)
   (* (/ (- y z) (- a z)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1.05e+156) || !(a <= 4.1e+102)) {
		tmp = fma((y - z), (t / a), x);
	} else {
		tmp = ((y - z) / (a - z)) * t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -1.05e+156) || !(a <= 4.1e+102))
		tmp = fma(Float64(y - z), Float64(t / a), x);
	else
		tmp = Float64(Float64(Float64(y - z) / Float64(a - z)) * t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -1.05e+156], N[Not[LessEqual[a, 4.1e+102]], $MachinePrecision]], N[(N[(y - z), $MachinePrecision] * N[(t / a), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(y - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.04999999999999991e156 or 4.1e102 < a

   1. Initial program 73.1%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x + \frac{\left(t - x\right) \cdot \left(y - z\right)}{a}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(t - x\right) \cdot \left(y - z\right)}{a} + x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(y - z\right) \cdot \left(t - x\right)}}{a} + x \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a}} + x \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - z}, \frac{t - x}{a}, x\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y - z, \color{blue}{\frac{t - x}{a}}, x\right) \]

      7. lower--.f64 83.0

         \[\leadsto \mathsf{fma}\left(y - z, \frac{\color{blue}{t - x}}{a}, x\right) \]

   5. Applied rewrites 83.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(y - z, \frac{t}{\color{blue}{a}}, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 80.1%

      \[\leadsto \mathsf{fma}\left(y - z, \frac{t}{\color{blue}{a}}, x\right) \]

   if -1.04999999999999991e156 < a < 4.1e102

   1. Initial program 66.3%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(y - z\right) \cdot \left(t - x\right)}}{a - z} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t - x}{a - z} \cdot \left(y - z\right)} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

      8. lower-/.f64 71.6

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{t - x}{a - z}}, y - z, x\right) \]

   4. Applied rewrites 71.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(\left(-1 \cdot \frac{x \cdot \left(y - z\right)}{t \cdot \left(a - z\right)} + \left(\frac{x}{t} + \frac{y}{a - z}\right)\right) - \frac{z}{a - z}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot \frac{x \cdot \left(y - z\right)}{t \cdot \left(a - z\right)} + \left(\frac{x}{t} + \frac{y}{a - z}\right)\right) - \frac{z}{a - z}\right) \cdot t} \]

   7. Applied rewrites 77.5%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left(-x, \frac{y - z}{a - z}, x\right)}{t} + \frac{y - z}{a - z}\right) \cdot t} \]

   8. Taylor expanded in x around 0

      \[\leadsto \left(\frac{y}{a - z} - \frac{z}{a - z}\right) \cdot t \]
   9. Step-by-step derivation

   10. Applied rewrites 61.5%

       \[\leadsto \frac{y - z}{a - z} \cdot t \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.05 \cdot 10^{+156} \lor \neg \left(a \leq 4.1 \cdot 10^{+102}\right):\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z}{a - z} \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 61.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{+25} \lor \neg \left(z \leq 1.38 \cdot 10^{+104}\right):\\ \;\;\;\;\left(-t\right) \cdot \frac{z}{a - z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -8.2e+25) (not (<= z 1.38e+104)))
   (* (- t) (/ z (- a z)))
   (fma (/ (- t x) a) y x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -8.2e+25) || !(z <= 1.38e+104)) {
		tmp = -t * (z / (a - z));
	} else {
		tmp = fma(((t - x) / a), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -8.2e+25) || !(z <= 1.38e+104))
		tmp = Float64(Float64(-t) * Float64(z / Float64(a - z)));
	else
		tmp = fma(Float64(Float64(t - x) / a), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -8.2e+25], N[Not[LessEqual[z, 1.38e+104]], $MachinePrecision]], N[((-t) * N[(z / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.19999999999999933e25 or 1.38e104 < z

   1. Initial program 39.8%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(y - z\right) \cdot \left(t - x\right)}}{a - z} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t - x}{a - z} \cdot \left(y - z\right)} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

      8. lower-/.f64 62.1

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{t - x}{a - z}}, y - z, x\right) \]

   4. Applied rewrites 62.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + \left(-1 \cdot \frac{y - z}{a - z} + \frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + \left(-1 \cdot \frac{y - z}{a - z} + \frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)}\right)\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \left(-1 \cdot \frac{y - z}{a - z} + \frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)}\right)\right) \cdot x} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot \frac{y - z}{a - z} + \frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)}\right) + 1\right)} \cdot x \]

      4. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)} + -1 \cdot \frac{y - z}{a - z}\right)} + 1\right) \cdot x \]

      5. times-frac N/A

         \[\leadsto \left(\left(\color{blue}{\frac{t}{x} \cdot \frac{y - z}{a - z}} + -1 \cdot \frac{y - z}{a - z}\right) + 1\right) \cdot x \]

      6. distribute-rgt-out N/A

         \[\leadsto \left(\color{blue}{\frac{y - z}{a - z} \cdot \left(\frac{t}{x} + -1\right)} + 1\right) \cdot x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, \frac{t}{x} + -1, 1\right)} \cdot x \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, \frac{t}{x} + -1, 1\right) \cdot x \]

      9. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{y - z}}{a - z}, \frac{t}{x} + -1, 1\right) \cdot x \]

      10. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{\color{blue}{a - z}}, \frac{t}{x} + -1, 1\right) \cdot x \]

      11. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{a - z}, \color{blue}{\frac{t}{x} + -1}, 1\right) \cdot x \]

      12. lower-/.f64 54.8

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{a - z}, \color{blue}{\frac{t}{x}} + -1, 1\right) \cdot x \]

   7. Applied rewrites 54.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, \frac{t}{x} + -1, 1\right) \cdot x} \]

   8. Taylor expanded in t around inf

      \[\leadsto t \cdot \color{blue}{\left(x \cdot \left(\frac{y}{x \cdot \left(a - z\right)} - \frac{z}{x \cdot \left(a - z\right)}\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 32.9%

       \[\leadsto \left(t \cdot x\right) \cdot \color{blue}{\frac{\frac{y - z}{x}}{a - z}} \]

   11. Taylor expanded in y around 0

       \[\leadsto -1 \cdot \frac{t \cdot z}{\color{blue}{a - z}} \]
   12. Step-by-step derivation

   13. Applied rewrites 60.1%

       \[\leadsto \left(-t\right) \cdot \frac{z}{\color{blue}{a - z}} \]

   if -8.19999999999999933e25 < z < 1.38e104

   1. Initial program 88.6%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{y \cdot \left(t - x\right)}{a}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(t - x\right)}{a} + x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{t - x}{a}} + x \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t - x}{a} \cdot y} + x \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{t - x}{a}}, y, x\right) \]

      6. lower--.f64 57.9

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{t - x}}{a}, y, x\right) \]

   5. Applied rewrites 57.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{+25} \lor \neg \left(z \leq 1.38 \cdot 10^{+104}\right):\\ \;\;\;\;\left(-t\right) \cdot \frac{z}{a - z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 59.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -8.6 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t}{a}, x\right)\\ \mathbf{elif}\;a \leq 1.75 \cdot 10^{-9}:\\ \;\;\;\;\left(-t\right) \cdot \frac{y - z}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -8.6e-20)
   (fma (- y z) (/ t a) x)
   (if (<= a 1.75e-9) (* (- t) (/ (- y z) z)) (fma (/ (- t x) a) y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -8.6e-20) {
		tmp = fma((y - z), (t / a), x);
	} else if (a <= 1.75e-9) {
		tmp = -t * ((y - z) / z);
	} else {
		tmp = fma(((t - x) / a), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -8.6e-20)
		tmp = fma(Float64(y - z), Float64(t / a), x);
	elseif (a <= 1.75e-9)
		tmp = Float64(Float64(-t) * Float64(Float64(y - z) / z));
	else
		tmp = fma(Float64(Float64(t - x) / a), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -8.6e-20], N[(N[(y - z), $MachinePrecision] * N[(t / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[a, 1.75e-9], N[((-t) * N[(N[(y - z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] * y + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -8.60000000000000022e-20

   1. Initial program 71.2%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x + \frac{\left(t - x\right) \cdot \left(y - z\right)}{a}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(t - x\right) \cdot \left(y - z\right)}{a} + x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(y - z\right) \cdot \left(t - x\right)}}{a} + x \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a}} + x \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - z}, \frac{t - x}{a}, x\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y - z, \color{blue}{\frac{t - x}{a}}, x\right) \]

      7. lower--.f64 70.5

         \[\leadsto \mathsf{fma}\left(y - z, \frac{\color{blue}{t - x}}{a}, x\right) \]

   5. Applied rewrites 70.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(y - z, \frac{t}{\color{blue}{a}}, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 62.1%

      \[\leadsto \mathsf{fma}\left(y - z, \frac{t}{\color{blue}{a}}, x\right) \]

   if -8.60000000000000022e-20 < a < 1.75e-9

   1. Initial program 67.2%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(y - z\right) \cdot \left(t - x\right)}}{a - z} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t - x}{a - z} \cdot \left(y - z\right)} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

      8. lower-/.f64 69.7

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{t - x}{a - z}}, y - z, x\right) \]

   4. Applied rewrites 69.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + \left(-1 \cdot \frac{y - z}{a - z} + \frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + \left(-1 \cdot \frac{y - z}{a - z} + \frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)}\right)\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \left(-1 \cdot \frac{y - z}{a - z} + \frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)}\right)\right) \cdot x} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot \frac{y - z}{a - z} + \frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)}\right) + 1\right)} \cdot x \]

      4. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)} + -1 \cdot \frac{y - z}{a - z}\right)} + 1\right) \cdot x \]

      5. times-frac N/A

         \[\leadsto \left(\left(\color{blue}{\frac{t}{x} \cdot \frac{y - z}{a - z}} + -1 \cdot \frac{y - z}{a - z}\right) + 1\right) \cdot x \]

      6. distribute-rgt-out N/A

         \[\leadsto \left(\color{blue}{\frac{y - z}{a - z} \cdot \left(\frac{t}{x} + -1\right)} + 1\right) \cdot x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, \frac{t}{x} + -1, 1\right)} \cdot x \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, \frac{t}{x} + -1, 1\right) \cdot x \]

      9. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{y - z}}{a - z}, \frac{t}{x} + -1, 1\right) \cdot x \]

      10. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{\color{blue}{a - z}}, \frac{t}{x} + -1, 1\right) \cdot x \]

      11. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{a - z}, \color{blue}{\frac{t}{x} + -1}, 1\right) \cdot x \]

      12. lower-/.f64 64.5

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{a - z}, \color{blue}{\frac{t}{x}} + -1, 1\right) \cdot x \]

   7. Applied rewrites 64.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, \frac{t}{x} + -1, 1\right) \cdot x} \]

   8. Taylor expanded in t around inf

      \[\leadsto t \cdot \color{blue}{\left(x \cdot \left(\frac{y}{x \cdot \left(a - z\right)} - \frac{z}{x \cdot \left(a - z\right)}\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 41.8%

       \[\leadsto \left(t \cdot x\right) \cdot \color{blue}{\frac{\frac{y - z}{x}}{a - z}} \]

   11. Taylor expanded in a around 0

       \[\leadsto -1 \cdot \frac{t \cdot \left(y - z\right)}{\color{blue}{z}} \]
   12. Step-by-step derivation

   13. Applied rewrites 58.7%

       \[\leadsto \left(-t\right) \cdot \frac{y - z}{\color{blue}{z}} \]

   if 1.75e-9 < a

   1. Initial program 66.8%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{y \cdot \left(t - x\right)}{a}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(t - x\right)}{a} + x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{t - x}{a}} + x \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t - x}{a} \cdot y} + x \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{t - x}{a}}, y, x\right) \]

      6. lower--.f64 63.6

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{t - x}}{a}, y, x\right) \]

   5. Applied rewrites 63.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -8.6 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t}{a}, x\right)\\ \mathbf{elif}\;a \leq 1.75 \cdot 10^{-9}:\\ \;\;\;\;\left(-t\right) \cdot \frac{y - z}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 56.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+38} \lor \neg \left(z \leq 3.1 \cdot 10^{+105}\right):\\ \;\;\;\;\frac{t}{x} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -1.5e+38) (not (<= z 3.1e+105)))
   (* (/ t x) x)
   (fma (/ (- t x) a) y x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.5e+38) || !(z <= 3.1e+105)) {
		tmp = (t / x) * x;
	} else {
		tmp = fma(((t - x) / a), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -1.5e+38) || !(z <= 3.1e+105))
		tmp = Float64(Float64(t / x) * x);
	else
		tmp = fma(Float64(Float64(t - x) / a), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -1.5e+38], N[Not[LessEqual[z, 3.1e+105]], $MachinePrecision]], N[(N[(t / x), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.5000000000000001e38 or 3.10000000000000004e105 < z

   1. Initial program 39.2%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(y - z\right) \cdot \left(t - x\right)}}{a - z} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t - x}{a - z} \cdot \left(y - z\right)} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

      8. lower-/.f64 61.8

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{t - x}{a - z}}, y - z, x\right) \]

   4. Applied rewrites 61.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + \left(-1 \cdot \frac{y - z}{a - z} + \frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + \left(-1 \cdot \frac{y - z}{a - z} + \frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)}\right)\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \left(-1 \cdot \frac{y - z}{a - z} + \frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)}\right)\right) \cdot x} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot \frac{y - z}{a - z} + \frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)}\right) + 1\right)} \cdot x \]

      4. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)} + -1 \cdot \frac{y - z}{a - z}\right)} + 1\right) \cdot x \]

      5. times-frac N/A

         \[\leadsto \left(\left(\color{blue}{\frac{t}{x} \cdot \frac{y - z}{a - z}} + -1 \cdot \frac{y - z}{a - z}\right) + 1\right) \cdot x \]

      6. distribute-rgt-out N/A

         \[\leadsto \left(\color{blue}{\frac{y - z}{a - z} \cdot \left(\frac{t}{x} + -1\right)} + 1\right) \cdot x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, \frac{t}{x} + -1, 1\right)} \cdot x \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, \frac{t}{x} + -1, 1\right) \cdot x \]

      9. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{y - z}}{a - z}, \frac{t}{x} + -1, 1\right) \cdot x \]

      10. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{\color{blue}{a - z}}, \frac{t}{x} + -1, 1\right) \cdot x \]

      11. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{a - z}, \color{blue}{\frac{t}{x} + -1}, 1\right) \cdot x \]

      12. lower-/.f64 55.2

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{a - z}, \color{blue}{\frac{t}{x}} + -1, 1\right) \cdot x \]

   7. Applied rewrites 55.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, \frac{t}{x} + -1, 1\right) \cdot x} \]

   8. Taylor expanded in z around inf

      \[\leadsto \frac{t}{x} \cdot x \]
   9. Step-by-step derivation

   10. Applied rewrites 47.4%

       \[\leadsto \frac{t}{x} \cdot x \]

   if -1.5000000000000001e38 < z < 3.10000000000000004e105

   1. Initial program 88.6%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{y \cdot \left(t - x\right)}{a}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(t - x\right)}{a} + x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{t - x}{a}} + x \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t - x}{a} \cdot y} + x \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{t - x}{a}}, y, x\right) \]

      6. lower--.f64 57.5

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{t - x}}{a}, y, x\right) \]

   5. Applied rewrites 57.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+38} \lor \neg \left(z \leq 3.1 \cdot 10^{+105}\right):\\ \;\;\;\;\frac{t}{x} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 51.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+26} \lor \neg \left(z \leq 3.1 \cdot 10^{+105}\right):\\ \;\;\;\;\frac{t}{x} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -1.9e+26) (not (<= z 3.1e+105)))
   (* (/ t x) x)
   (fma (- y z) (/ t a) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.9e+26) || !(z <= 3.1e+105)) {
		tmp = (t / x) * x;
	} else {
		tmp = fma((y - z), (t / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -1.9e+26) || !(z <= 3.1e+105))
		tmp = Float64(Float64(t / x) * x);
	else
		tmp = fma(Float64(y - z), Float64(t / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -1.9e+26], N[Not[LessEqual[z, 3.1e+105]], $MachinePrecision]], N[(N[(t / x), $MachinePrecision] * x), $MachinePrecision], N[(N[(y - z), $MachinePrecision] * N[(t / a), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.9000000000000001e26 or 3.10000000000000004e105 < z

   1. Initial program 39.8%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(y - z\right) \cdot \left(t - x\right)}}{a - z} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t - x}{a - z} \cdot \left(y - z\right)} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

      8. lower-/.f64 62.1

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{t - x}{a - z}}, y - z, x\right) \]

   4. Applied rewrites 62.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + \left(-1 \cdot \frac{y - z}{a - z} + \frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + \left(-1 \cdot \frac{y - z}{a - z} + \frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)}\right)\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \left(-1 \cdot \frac{y - z}{a - z} + \frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)}\right)\right) \cdot x} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot \frac{y - z}{a - z} + \frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)}\right) + 1\right)} \cdot x \]

      4. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)} + -1 \cdot \frac{y - z}{a - z}\right)} + 1\right) \cdot x \]

      5. times-frac N/A

         \[\leadsto \left(\left(\color{blue}{\frac{t}{x} \cdot \frac{y - z}{a - z}} + -1 \cdot \frac{y - z}{a - z}\right) + 1\right) \cdot x \]

      6. distribute-rgt-out N/A

         \[\leadsto \left(\color{blue}{\frac{y - z}{a - z} \cdot \left(\frac{t}{x} + -1\right)} + 1\right) \cdot x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, \frac{t}{x} + -1, 1\right)} \cdot x \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, \frac{t}{x} + -1, 1\right) \cdot x \]

      9. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{y - z}}{a - z}, \frac{t}{x} + -1, 1\right) \cdot x \]

      10. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{\color{blue}{a - z}}, \frac{t}{x} + -1, 1\right) \cdot x \]

      11. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{a - z}, \color{blue}{\frac{t}{x} + -1}, 1\right) \cdot x \]

      12. lower-/.f64 54.8

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{a - z}, \color{blue}{\frac{t}{x}} + -1, 1\right) \cdot x \]

   7. Applied rewrites 54.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, \frac{t}{x} + -1, 1\right) \cdot x} \]

   8. Taylor expanded in z around inf

      \[\leadsto \frac{t}{x} \cdot x \]
   9. Step-by-step derivation

   10. Applied rewrites 47.0%

       \[\leadsto \frac{t}{x} \cdot x \]

   if -1.9000000000000001e26 < z < 3.10000000000000004e105

   1. Initial program 88.6%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x + \frac{\left(t - x\right) \cdot \left(y - z\right)}{a}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(t - x\right) \cdot \left(y - z\right)}{a} + x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(y - z\right) \cdot \left(t - x\right)}}{a} + x \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a}} + x \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - z}, \frac{t - x}{a}, x\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y - z, \color{blue}{\frac{t - x}{a}}, x\right) \]

      7. lower--.f64 62.5

         \[\leadsto \mathsf{fma}\left(y - z, \frac{\color{blue}{t - x}}{a}, x\right) \]

   5. Applied rewrites 62.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(y - z, \frac{t}{\color{blue}{a}}, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 51.2%

      \[\leadsto \mathsf{fma}\left(y - z, \frac{t}{\color{blue}{a}}, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+26} \lor \neg \left(z \leq 3.1 \cdot 10^{+105}\right):\\ \;\;\;\;\frac{t}{x} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 45.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.9 \cdot 10^{+14} \lor \neg \left(z \leq 2.65 \cdot 10^{+105}\right):\\ \;\;\;\;\frac{t}{x} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-x, \frac{y}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -3.9e+14) (not (<= z 2.65e+105)))
   (* (/ t x) x)
   (fma (- x) (/ y a) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -3.9e+14) || !(z <= 2.65e+105)) {
		tmp = (t / x) * x;
	} else {
		tmp = fma(-x, (y / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -3.9e+14) || !(z <= 2.65e+105))
		tmp = Float64(Float64(t / x) * x);
	else
		tmp = fma(Float64(-x), Float64(y / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -3.9e+14], N[Not[LessEqual[z, 2.65e+105]], $MachinePrecision]], N[(N[(t / x), $MachinePrecision] * x), $MachinePrecision], N[((-x) * N[(y / a), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.9e14 or 2.6499999999999999e105 < z

   1. Initial program 40.3%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(y - z\right) \cdot \left(t - x\right)}}{a - z} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t - x}{a - z} \cdot \left(y - z\right)} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

      8. lower-/.f64 62.5

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{t - x}{a - z}}, y - z, x\right) \]

   4. Applied rewrites 62.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + \left(-1 \cdot \frac{y - z}{a - z} + \frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + \left(-1 \cdot \frac{y - z}{a - z} + \frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)}\right)\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \left(-1 \cdot \frac{y - z}{a - z} + \frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)}\right)\right) \cdot x} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot \frac{y - z}{a - z} + \frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)}\right) + 1\right)} \cdot x \]

      4. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)} + -1 \cdot \frac{y - z}{a - z}\right)} + 1\right) \cdot x \]

      5. times-frac N/A

         \[\leadsto \left(\left(\color{blue}{\frac{t}{x} \cdot \frac{y - z}{a - z}} + -1 \cdot \frac{y - z}{a - z}\right) + 1\right) \cdot x \]

      6. distribute-rgt-out N/A

         \[\leadsto \left(\color{blue}{\frac{y - z}{a - z} \cdot \left(\frac{t}{x} + -1\right)} + 1\right) \cdot x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, \frac{t}{x} + -1, 1\right)} \cdot x \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, \frac{t}{x} + -1, 1\right) \cdot x \]

      9. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{y - z}}{a - z}, \frac{t}{x} + -1, 1\right) \cdot x \]

      10. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{\color{blue}{a - z}}, \frac{t}{x} + -1, 1\right) \cdot x \]

      11. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{a - z}, \color{blue}{\frac{t}{x} + -1}, 1\right) \cdot x \]

      12. lower-/.f64 55.2

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{a - z}, \color{blue}{\frac{t}{x}} + -1, 1\right) \cdot x \]

   7. Applied rewrites 55.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, \frac{t}{x} + -1, 1\right) \cdot x} \]

   8. Taylor expanded in z around inf

      \[\leadsto \frac{t}{x} \cdot x \]
   9. Step-by-step derivation

   10. Applied rewrites 47.5%

       \[\leadsto \frac{t}{x} \cdot x \]

   if -3.9e14 < z < 2.6499999999999999e105

   1. Initial program 88.5%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(y - z\right) \cdot \left(t - x\right)}}{a - z} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t - x}{a - z} \cdot \left(y - z\right)} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

      8. lower-/.f64 86.8

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{t - x}{a - z}}, y - z, x\right) \]

   4. Applied rewrites 86.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + -1 \cdot \frac{y - z}{a - z}\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \frac{y - z}{a - z} + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \frac{y - z}{a - z}\right) + x \cdot 1} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{y - z}{a - z}\right)\right)} + x \cdot 1 \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{y - z}{a - z}\right)\right)} + x \cdot 1 \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{y - z}{a - z}} + x \cdot 1 \]

      6. *-rgt-identity N/A

         \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \frac{y - z}{a - z} + \color{blue}{x} \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{y - z}{a - z}, x\right)} \]

      8. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-x}, \frac{y - z}{a - z}, x\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(-x, \color{blue}{\frac{y - z}{a - z}}, x\right) \]

      10. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(-x, \frac{\color{blue}{y - z}}{a - z}, x\right) \]

      11. lower--.f64 53.3

          \[\leadsto \mathsf{fma}\left(-x, \frac{y - z}{\color{blue}{a - z}}, x\right) \]

   7. Applied rewrites 53.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-x, \frac{y - z}{a - z}, x\right)} \]

   8. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(-x, \frac{y}{\color{blue}{a}}, x\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 45.6%

       \[\leadsto \mathsf{fma}\left(-x, \frac{y}{\color{blue}{a}}, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.9 \cdot 10^{+14} \lor \neg \left(z \leq 2.65 \cdot 10^{+105}\right):\\ \;\;\;\;\frac{t}{x} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-x, \frac{y}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 34.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.6 \cdot 10^{+136} \lor \neg \left(a \leq 2.7 \cdot 10^{+102}\right):\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{x} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -2.6e+136) (not (<= a 2.7e+102))) (* 1.0 x) (* (/ t x) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -2.6e+136) || !(a <= 2.7e+102)) {
		tmp = 1.0 * x;
	} else {
		tmp = (t / x) * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-2.6d+136)) .or. (.not. (a <= 2.7d+102))) then
        tmp = 1.0d0 * x
    else
        tmp = (t / x) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -2.6e+136) || !(a <= 2.7e+102)) {
		tmp = 1.0 * x;
	} else {
		tmp = (t / x) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -2.6e+136) or not (a <= 2.7e+102):
		tmp = 1.0 * x
	else:
		tmp = (t / x) * x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -2.6e+136) || !(a <= 2.7e+102))
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(Float64(t / x) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -2.6e+136) || ~((a <= 2.7e+102)))
		tmp = 1.0 * x;
	else
		tmp = (t / x) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -2.6e+136], N[Not[LessEqual[a, 2.7e+102]], $MachinePrecision]], N[(1.0 * x), $MachinePrecision], N[(N[(t / x), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.6000000000000001e136 or 2.7000000000000001e102 < a

   1. Initial program 73.9%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(y - z\right) \cdot \left(t - x\right)}}{a - z} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t - x}{a - z} \cdot \left(y - z\right)} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

      8. lower-/.f64 89.8

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{t - x}{a - z}}, y - z, x\right) \]

   4. Applied rewrites 89.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + \left(-1 \cdot \frac{y - z}{a - z} + \frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + \left(-1 \cdot \frac{y - z}{a - z} + \frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)}\right)\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \left(-1 \cdot \frac{y - z}{a - z} + \frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)}\right)\right) \cdot x} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot \frac{y - z}{a - z} + \frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)}\right) + 1\right)} \cdot x \]

      4. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)} + -1 \cdot \frac{y - z}{a - z}\right)} + 1\right) \cdot x \]

      5. times-frac N/A

         \[\leadsto \left(\left(\color{blue}{\frac{t}{x} \cdot \frac{y - z}{a - z}} + -1 \cdot \frac{y - z}{a - z}\right) + 1\right) \cdot x \]

      6. distribute-rgt-out N/A

         \[\leadsto \left(\color{blue}{\frac{y - z}{a - z} \cdot \left(\frac{t}{x} + -1\right)} + 1\right) \cdot x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, \frac{t}{x} + -1, 1\right)} \cdot x \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, \frac{t}{x} + -1, 1\right) \cdot x \]

      9. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{y - z}}{a - z}, \frac{t}{x} + -1, 1\right) \cdot x \]

      10. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{\color{blue}{a - z}}, \frac{t}{x} + -1, 1\right) \cdot x \]

      11. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{a - z}, \color{blue}{\frac{t}{x} + -1}, 1\right) \cdot x \]

      12. lower-/.f64 77.9

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{a - z}, \color{blue}{\frac{t}{x}} + -1, 1\right) \cdot x \]

   7. Applied rewrites 77.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, \frac{t}{x} + -1, 1\right) \cdot x} \]

   8. Taylor expanded in z around inf

      \[\leadsto \frac{t}{x} \cdot x \]
   9. Step-by-step derivation

   10. Applied rewrites 6.4%

       \[\leadsto \frac{t}{x} \cdot x \]

   11. Taylor expanded in a around inf

       \[\leadsto 1 \cdot x \]
   12. Step-by-step derivation

   13. Applied rewrites 60.4%

       \[\leadsto 1 \cdot x \]

   if -2.6000000000000001e136 < a < 2.7000000000000001e102

   1. Initial program 65.9%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(y - z\right) \cdot \left(t - x\right)}}{a - z} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t - x}{a - z} \cdot \left(y - z\right)} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

      8. lower-/.f64 71.8

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{t - x}{a - z}}, y - z, x\right) \]

   4. Applied rewrites 71.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + \left(-1 \cdot \frac{y - z}{a - z} + \frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + \left(-1 \cdot \frac{y - z}{a - z} + \frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)}\right)\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \left(-1 \cdot \frac{y - z}{a - z} + \frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)}\right)\right) \cdot x} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot \frac{y - z}{a - z} + \frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)}\right) + 1\right)} \cdot x \]

      4. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)} + -1 \cdot \frac{y - z}{a - z}\right)} + 1\right) \cdot x \]

      5. times-frac N/A

         \[\leadsto \left(\left(\color{blue}{\frac{t}{x} \cdot \frac{y - z}{a - z}} + -1 \cdot \frac{y - z}{a - z}\right) + 1\right) \cdot x \]

      6. distribute-rgt-out N/A

         \[\leadsto \left(\color{blue}{\frac{y - z}{a - z} \cdot \left(\frac{t}{x} + -1\right)} + 1\right) \cdot x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, \frac{t}{x} + -1, 1\right)} \cdot x \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, \frac{t}{x} + -1, 1\right) \cdot x \]

      9. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{y - z}}{a - z}, \frac{t}{x} + -1, 1\right) \cdot x \]

      10. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{\color{blue}{a - z}}, \frac{t}{x} + -1, 1\right) \cdot x \]

      11. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{a - z}, \color{blue}{\frac{t}{x} + -1}, 1\right) \cdot x \]

      12. lower-/.f64 66.4

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{a - z}, \color{blue}{\frac{t}{x}} + -1, 1\right) \cdot x \]

   7. Applied rewrites 66.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, \frac{t}{x} + -1, 1\right) \cdot x} \]

   8. Taylor expanded in z around inf

      \[\leadsto \frac{t}{x} \cdot x \]
   9. Step-by-step derivation

   10. Applied rewrites 36.9%

       \[\leadsto \frac{t}{x} \cdot x \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.6 \cdot 10^{+136} \lor \neg \left(a \leq 2.7 \cdot 10^{+102}\right):\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{x} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 34.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.6 \cdot 10^{+136}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{z}{a}, x\right)\\ \mathbf{elif}\;a \leq 2.7 \cdot 10^{+102}:\\ \;\;\;\;\frac{t}{x} \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -2.6e+136)
   (fma x (/ z a) x)
   (if (<= a 2.7e+102) (* (/ t x) x) (* 1.0 x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -2.6e+136) {
		tmp = fma(x, (z / a), x);
	} else if (a <= 2.7e+102) {
		tmp = (t / x) * x;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -2.6e+136)
		tmp = fma(x, Float64(z / a), x);
	elseif (a <= 2.7e+102)
		tmp = Float64(Float64(t / x) * x);
	else
		tmp = Float64(1.0 * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -2.6e+136], N[(x * N[(z / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[a, 2.7e+102], N[(N[(t / x), $MachinePrecision] * x), $MachinePrecision], N[(1.0 * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.6000000000000001e136

   1. Initial program 78.7%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(y - z\right) \cdot \left(t - x\right)}}{a - z} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t - x}{a - z} \cdot \left(y - z\right)} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

      8. lower-/.f64 87.9

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{t - x}{a - z}}, y - z, x\right) \]

   4. Applied rewrites 87.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + -1 \cdot \frac{y - z}{a - z}\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \frac{y - z}{a - z} + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \frac{y - z}{a - z}\right) + x \cdot 1} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{y - z}{a - z}\right)\right)} + x \cdot 1 \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{y - z}{a - z}\right)\right)} + x \cdot 1 \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{y - z}{a - z}} + x \cdot 1 \]

      6. *-rgt-identity N/A

         \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \frac{y - z}{a - z} + \color{blue}{x} \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{y - z}{a - z}, x\right)} \]

      8. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-x}, \frac{y - z}{a - z}, x\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(-x, \color{blue}{\frac{y - z}{a - z}}, x\right) \]

      10. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(-x, \frac{\color{blue}{y - z}}{a - z}, x\right) \]

      11. lower--.f64 58.0

          \[\leadsto \mathsf{fma}\left(-x, \frac{y - z}{\color{blue}{a - z}}, x\right) \]

   7. Applied rewrites 58.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-x, \frac{y - z}{a - z}, x\right)} \]

   8. Taylor expanded in y around 0

      \[\leadsto x + \color{blue}{\frac{x \cdot z}{a - z}} \]
   9. Step-by-step derivation

   10. Applied rewrites 54.8%

       \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{z}{a - z}}, x\right) \]

   11. Taylor expanded in z around 0

       \[\leadsto \mathsf{fma}\left(x, \frac{z}{a}, x\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 54.9%

       \[\leadsto \mathsf{fma}\left(x, \frac{z}{a}, x\right) \]

   if -2.6000000000000001e136 < a < 2.7000000000000001e102

   1. Initial program 65.9%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(y - z\right) \cdot \left(t - x\right)}}{a - z} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t - x}{a - z} \cdot \left(y - z\right)} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

      8. lower-/.f64 71.8

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{t - x}{a - z}}, y - z, x\right) \]

   4. Applied rewrites 71.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + \left(-1 \cdot \frac{y - z}{a - z} + \frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + \left(-1 \cdot \frac{y - z}{a - z} + \frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)}\right)\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \left(-1 \cdot \frac{y - z}{a - z} + \frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)}\right)\right) \cdot x} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot \frac{y - z}{a - z} + \frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)}\right) + 1\right)} \cdot x \]

      4. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)} + -1 \cdot \frac{y - z}{a - z}\right)} + 1\right) \cdot x \]

      5. times-frac N/A

         \[\leadsto \left(\left(\color{blue}{\frac{t}{x} \cdot \frac{y - z}{a - z}} + -1 \cdot \frac{y - z}{a - z}\right) + 1\right) \cdot x \]

      6. distribute-rgt-out N/A

         \[\leadsto \left(\color{blue}{\frac{y - z}{a - z} \cdot \left(\frac{t}{x} + -1\right)} + 1\right) \cdot x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, \frac{t}{x} + -1, 1\right)} \cdot x \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, \frac{t}{x} + -1, 1\right) \cdot x \]

      9. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{y - z}}{a - z}, \frac{t}{x} + -1, 1\right) \cdot x \]

      10. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{\color{blue}{a - z}}, \frac{t}{x} + -1, 1\right) \cdot x \]

      11. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{a - z}, \color{blue}{\frac{t}{x} + -1}, 1\right) \cdot x \]

      12. lower-/.f64 66.4

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{a - z}, \color{blue}{\frac{t}{x}} + -1, 1\right) \cdot x \]

   7. Applied rewrites 66.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, \frac{t}{x} + -1, 1\right) \cdot x} \]

   8. Taylor expanded in z around inf

      \[\leadsto \frac{t}{x} \cdot x \]
   9. Step-by-step derivation

   10. Applied rewrites 36.9%

       \[\leadsto \frac{t}{x} \cdot x \]

   if 2.7000000000000001e102 < a

   1. Initial program 69.4%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(y - z\right) \cdot \left(t - x\right)}}{a - z} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t - x}{a - z} \cdot \left(y - z\right)} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

      8. lower-/.f64 91.5

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{t - x}{a - z}}, y - z, x\right) \]

   4. Applied rewrites 91.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + \left(-1 \cdot \frac{y - z}{a - z} + \frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + \left(-1 \cdot \frac{y - z}{a - z} + \frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)}\right)\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \left(-1 \cdot \frac{y - z}{a - z} + \frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)}\right)\right) \cdot x} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot \frac{y - z}{a - z} + \frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)}\right) + 1\right)} \cdot x \]

      4. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)} + -1 \cdot \frac{y - z}{a - z}\right)} + 1\right) \cdot x \]

      5. times-frac N/A

         \[\leadsto \left(\left(\color{blue}{\frac{t}{x} \cdot \frac{y - z}{a - z}} + -1 \cdot \frac{y - z}{a - z}\right) + 1\right) \cdot x \]

      6. distribute-rgt-out N/A

         \[\leadsto \left(\color{blue}{\frac{y - z}{a - z} \cdot \left(\frac{t}{x} + -1\right)} + 1\right) \cdot x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, \frac{t}{x} + -1, 1\right)} \cdot x \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, \frac{t}{x} + -1, 1\right) \cdot x \]

      9. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{y - z}}{a - z}, \frac{t}{x} + -1, 1\right) \cdot x \]

      10. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{\color{blue}{a - z}}, \frac{t}{x} + -1, 1\right) \cdot x \]

      11. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{a - z}, \color{blue}{\frac{t}{x} + -1}, 1\right) \cdot x \]

      12. lower-/.f64 85.7

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{a - z}, \color{blue}{\frac{t}{x}} + -1, 1\right) \cdot x \]

   7. Applied rewrites 85.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, \frac{t}{x} + -1, 1\right) \cdot x} \]

   8. Taylor expanded in z around inf

      \[\leadsto \frac{t}{x} \cdot x \]
   9. Step-by-step derivation

   10. Applied rewrites 6.1%

       \[\leadsto \frac{t}{x} \cdot x \]

   11. Taylor expanded in a around inf

       \[\leadsto 1 \cdot x \]
   12. Step-by-step derivation

   13. Applied rewrites 65.8%

       \[\leadsto 1 \cdot x \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 21: 31.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.9 \cdot 10^{+135} \lor \neg \left(a \leq 1.75 \cdot 10^{+102}\right):\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;x + \left(t - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -1.9e+135) (not (<= a 1.75e+102))) (* 1.0 x) (+ x (- t x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1.9e+135) || !(a <= 1.75e+102)) {
		tmp = 1.0 * x;
	} else {
		tmp = x + (t - x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-1.9d+135)) .or. (.not. (a <= 1.75d+102))) then
        tmp = 1.0d0 * x
    else
        tmp = x + (t - x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1.9e+135) || !(a <= 1.75e+102)) {
		tmp = 1.0 * x;
	} else {
		tmp = x + (t - x);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -1.9e+135) or not (a <= 1.75e+102):
		tmp = 1.0 * x
	else:
		tmp = x + (t - x)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -1.9e+135) || !(a <= 1.75e+102))
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(x + Float64(t - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -1.9e+135) || ~((a <= 1.75e+102)))
		tmp = 1.0 * x;
	else
		tmp = x + (t - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -1.9e+135], N[Not[LessEqual[a, 1.75e+102]], $MachinePrecision]], N[(1.0 * x), $MachinePrecision], N[(x + N[(t - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.9000000000000001e135 or 1.75000000000000005e102 < a

   1. Initial program 73.9%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(y - z\right) \cdot \left(t - x\right)}}{a - z} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t - x}{a - z} \cdot \left(y - z\right)} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

      8. lower-/.f64 89.8

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{t - x}{a - z}}, y - z, x\right) \]

   4. Applied rewrites 89.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + \left(-1 \cdot \frac{y - z}{a - z} + \frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + \left(-1 \cdot \frac{y - z}{a - z} + \frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)}\right)\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \left(-1 \cdot \frac{y - z}{a - z} + \frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)}\right)\right) \cdot x} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot \frac{y - z}{a - z} + \frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)}\right) + 1\right)} \cdot x \]

      4. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)} + -1 \cdot \frac{y - z}{a - z}\right)} + 1\right) \cdot x \]

      5. times-frac N/A

         \[\leadsto \left(\left(\color{blue}{\frac{t}{x} \cdot \frac{y - z}{a - z}} + -1 \cdot \frac{y - z}{a - z}\right) + 1\right) \cdot x \]

      6. distribute-rgt-out N/A

         \[\leadsto \left(\color{blue}{\frac{y - z}{a - z} \cdot \left(\frac{t}{x} + -1\right)} + 1\right) \cdot x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, \frac{t}{x} + -1, 1\right)} \cdot x \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, \frac{t}{x} + -1, 1\right) \cdot x \]

      9. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{y - z}}{a - z}, \frac{t}{x} + -1, 1\right) \cdot x \]

      10. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{\color{blue}{a - z}}, \frac{t}{x} + -1, 1\right) \cdot x \]

      11. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{a - z}, \color{blue}{\frac{t}{x} + -1}, 1\right) \cdot x \]

      12. lower-/.f64 77.9

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{a - z}, \color{blue}{\frac{t}{x}} + -1, 1\right) \cdot x \]

   7. Applied rewrites 77.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, \frac{t}{x} + -1, 1\right) \cdot x} \]

   8. Taylor expanded in z around inf

      \[\leadsto \frac{t}{x} \cdot x \]
   9. Step-by-step derivation

   10. Applied rewrites 6.4%

       \[\leadsto \frac{t}{x} \cdot x \]

   11. Taylor expanded in a around inf

       \[\leadsto 1 \cdot x \]
   12. Step-by-step derivation

   13. Applied rewrites 60.4%

       \[\leadsto 1 \cdot x \]

   if -1.9000000000000001e135 < a < 1.75000000000000005e102

   1. Initial program 65.9%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto x + \color{blue}{\left(t - x\right)} \]
   4. Step-by-step derivation

      1. lower--.f64 28.1

         \[\leadsto x + \color{blue}{\left(t - x\right)} \]

   5. Applied rewrites 28.1%

      \[\leadsto x + \color{blue}{\left(t - x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.9 \cdot 10^{+135} \lor \neg \left(a \leq 1.75 \cdot 10^{+102}\right):\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;x + \left(t - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 24.5% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ 1 \cdot x \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (* 1.0 x))

C

double code(double x, double y, double z, double t, double a) {
	return 1.0 * x;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = 1.0d0 * x
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return 1.0 * x;
}

Python

def code(x, y, z, t, a):
	return 1.0 * x

Julia

function code(x, y, z, t, a)
	return Float64(1.0 * x)
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = 1.0 * x;
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(1.0 * x), $MachinePrecision]

Derivation

1. Initial program 68.0%

   \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} + x} \]

   3. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} + x \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(y - z\right) \cdot \left(t - x\right)}}{a - z} + x \]

   5. associate-/l* N/A

      \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\frac{t - x}{a - z} \cdot \left(y - z\right)} + x \]

   7. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

   8. lower-/.f64 76.5

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{t - x}{a - z}}, y - z, x\right) \]

4. Applied rewrites 76.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

5. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x \cdot \left(1 + \left(-1 \cdot \frac{y - z}{a - z} + \frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)}\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(1 + \left(-1 \cdot \frac{y - z}{a - z} + \frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)}\right)\right) \cdot x} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(1 + \left(-1 \cdot \frac{y - z}{a - z} + \frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)}\right)\right) \cdot x} \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(-1 \cdot \frac{y - z}{a - z} + \frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)}\right) + 1\right)} \cdot x \]

   4. +-commutative N/A

      \[\leadsto \left(\color{blue}{\left(\frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)} + -1 \cdot \frac{y - z}{a - z}\right)} + 1\right) \cdot x \]

   5. times-frac N/A

      \[\leadsto \left(\left(\color{blue}{\frac{t}{x} \cdot \frac{y - z}{a - z}} + -1 \cdot \frac{y - z}{a - z}\right) + 1\right) \cdot x \]

   6. distribute-rgt-out N/A

      \[\leadsto \left(\color{blue}{\frac{y - z}{a - z} \cdot \left(\frac{t}{x} + -1\right)} + 1\right) \cdot x \]

   7. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, \frac{t}{x} + -1, 1\right)} \cdot x \]

   8. lower-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, \frac{t}{x} + -1, 1\right) \cdot x \]

   9. lower--.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{y - z}}{a - z}, \frac{t}{x} + -1, 1\right) \cdot x \]

   10. lower--.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{y - z}{\color{blue}{a - z}}, \frac{t}{x} + -1, 1\right) \cdot x \]

   11. lower-+.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{y - z}{a - z}, \color{blue}{\frac{t}{x} + -1}, 1\right) \cdot x \]

   12. lower-/.f64 69.4

       \[\leadsto \mathsf{fma}\left(\frac{y - z}{a - z}, \color{blue}{\frac{t}{x}} + -1, 1\right) \cdot x \]

7. Applied rewrites 69.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, \frac{t}{x} + -1, 1\right) \cdot x} \]

8. Taylor expanded in z around inf

   \[\leadsto \frac{t}{x} \cdot x \]
9. Step-by-step derivation

10. Applied rewrites 29.0%

    \[\leadsto \frac{t}{x} \cdot x \]

11. Taylor expanded in a around inf

    \[\leadsto 1 \cdot x \]
12. Step-by-step derivation

13. Applied rewrites 22.5%

    \[\leadsto 1 \cdot x \]
14. Add Preprocessing

Developer Target 1: 84.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - \frac{y}{z} \cdot \left(t - x\right)\\ \mathbf{if}\;z < -1.2536131056095036 \cdot 10^{+188}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z < 4.446702369113811 \cdot 10^{+64}:\\ \;\;\;\;x + \frac{y - z}{\frac{a - z}{t - x}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* (/ y z) (- t x)))))
   (if (< z -1.2536131056095036e+188)
     t_1
     (if (< z 4.446702369113811e+64)
       (+ x (/ (- y z) (/ (- a z) (- t x))))
       t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t - ((y / z) * (t - x));
	double tmp;
	if (z < -1.2536131056095036e+188) {
		tmp = t_1;
	} else if (z < 4.446702369113811e+64) {
		tmp = x + ((y - z) / ((a - z) / (t - x)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t - ((y / z) * (t - x))
    if (z < (-1.2536131056095036d+188)) then
        tmp = t_1
    else if (z < 4.446702369113811d+64) then
        tmp = x + ((y - z) / ((a - z) / (t - x)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = t - ((y / z) * (t - x));
	double tmp;
	if (z < -1.2536131056095036e+188) {
		tmp = t_1;
	} else if (z < 4.446702369113811e+64) {
		tmp = x + ((y - z) / ((a - z) / (t - x)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = t - ((y / z) * (t - x))
	tmp = 0
	if z < -1.2536131056095036e+188:
		tmp = t_1
	elif z < 4.446702369113811e+64:
		tmp = x + ((y - z) / ((a - z) / (t - x)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(Float64(y / z) * Float64(t - x)))
	tmp = 0.0
	if (z < -1.2536131056095036e+188)
		tmp = t_1;
	elseif (z < 4.446702369113811e+64)
		tmp = Float64(x + Float64(Float64(y - z) / Float64(Float64(a - z) / Float64(t - x))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = t - ((y / z) * (t - x));
	tmp = 0.0;
	if (z < -1.2536131056095036e+188)
		tmp = t_1;
	elseif (z < 4.446702369113811e+64)
		tmp = x + ((y - z) / ((a - z) / (t - x)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t - N[(N[(y / z), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[z, -1.2536131056095036e+188], t$95$1, If[Less[z, 4.446702369113811e+64], N[(x + N[(N[(y - z), $MachinePrecision] / N[(N[(a - z), $MachinePrecision] / N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2024320 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:invLinMap from Chart-1.5.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z -125361310560950360000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- t (* (/ y z) (- t x))) (if (< z 44467023691138110000000000000000000000000000000000000000000000000) (+ x (/ (- y z) (/ (- a z) (- t x)))) (- t (* (/ y z) (- t x))))))

  (+ x (/ (* (- y z) (- t x)) (- a z))))