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

Percentage Accurate: 67.5% → 89.4%

Time: 10.9s

Alternatives: 17

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: 89.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t - x, \frac{z - y}{z - a}, x\right)\\ t_2 := x - \frac{\left(z - y\right) \cdot \left(x - t\right)}{z - a}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-244}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;t - \frac{\left(a - y\right) \cdot \left(x - t\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- t x) (/ (- z y) (- z a)) x))
        (t_2 (- x (/ (* (- z y) (- x t)) (- z a)))))
   (if (<= t_2 -5e-244)
     t_1
     (if (<= t_2 0.0) (- t (/ (* (- a y) (- x t)) z)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((t - x), ((z - y) / (z - a)), x);
	double t_2 = x - (((z - y) * (x - t)) / (z - a));
	double tmp;
	if (t_2 <= -5e-244) {
		tmp = t_1;
	} else if (t_2 <= 0.0) {
		tmp = t - (((a - y) * (x - t)) / z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(t - x), Float64(Float64(z - y) / Float64(z - a)), x)
	t_2 = Float64(x - Float64(Float64(Float64(z - y) * Float64(x - t)) / Float64(z - a)))
	tmp = 0.0
	if (t_2 <= -5e-244)
		tmp = t_1;
	elseif (t_2 <= 0.0)
		tmp = Float64(t - Float64(Float64(Float64(a - y) * Float64(x - t)) / 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[(z - y), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(x - N[(N[(N[(z - y), $MachinePrecision] * N[(x - t), $MachinePrecision]), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e-244], t$95$1, If[LessEqual[t$95$2, 0.0], N[(t - N[(N[(N[(a - y), $MachinePrecision] * N[(x - t), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < -4.99999999999999998e-244 or 0.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z)))

   1. Initial program 74.9%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 92.3

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

   4. Applied rewrites 92.3%

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

   if -4.99999999999999998e-244 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < 0.0

   1. Initial program 7.1%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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. lift--.f64 N/A

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

      8. flip-- N/A

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

      9. associate-/r/ N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. difference-of-squares N/A

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

      14. lift--.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-+.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-+.f64 2.5

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

   4. Applied rewrites 2.5%

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

   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. 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. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. mul-1-neg N/A

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

      5. div-sub N/A

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

      6. mul-1-neg N/A

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

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

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

      8. associate-*r/ N/A

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

      9. mul-1-neg N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 N/A

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

   7. Applied rewrites 99.7%

      \[\leadsto \color{blue}{t - \frac{\left(t - x\right) \cdot \left(y - a\right)}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \frac{\left(z - y\right) \cdot \left(x - t\right)}{z - a} \leq -5 \cdot 10^{-244}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{z - y}{z - a}, x\right)\\ \mathbf{elif}\;x - \frac{\left(z - y\right) \cdot \left(x - t\right)}{z - a} \leq 0:\\ \;\;\;\;t - \frac{\left(a - y\right) \cdot \left(x - t\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{z - y}{z - a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{\mathsf{fma}\left(t, -1, x\right)}{z}, y - a, t\right)\\ \mathbf{if}\;z \leq -2.8 \cdot 10^{+63}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -2.8 \cdot 10^{-137}:\\ \;\;\;\;\frac{t}{z - a} \cdot \left(z - y\right) + x\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-54}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (fma t -1.0 x) z) (- y a) t)))
   (if (<= z -2.8e+63)
     t_1
     (if (<= z -2.8e-137)
       (+ (* (/ t (- z a)) (- z y)) x)
       (if (<= z 2.4e-54) (fma (/ (- y z) a) (- t x) x) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((fma(t, -1.0, x) / z), (y - a), t);
	double tmp;
	if (z <= -2.8e+63) {
		tmp = t_1;
	} else if (z <= -2.8e-137) {
		tmp = ((t / (z - a)) * (z - y)) + x;
	} else if (z <= 2.4e-54) {
		tmp = fma(((y - z) / a), (t - x), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(fma(t, -1.0, x) / z), Float64(y - a), t)
	tmp = 0.0
	if (z <= -2.8e+63)
		tmp = t_1;
	elseif (z <= -2.8e-137)
		tmp = Float64(Float64(Float64(t / Float64(z - a)) * Float64(z - y)) + x);
	elseif (z <= 2.4e-54)
		tmp = fma(Float64(Float64(y - z) / a), Float64(t - x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(t * -1.0 + x), $MachinePrecision] / z), $MachinePrecision] * N[(y - a), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -2.8e+63], t$95$1, If[LessEqual[z, -2.8e-137], N[(N[(N[(t / N[(z - a), $MachinePrecision]), $MachinePrecision] * N[(z - y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 2.4e-54], N[(N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.79999999999999987e63 or 2.40000000000000013e-54 < z

   1. Initial program 48.0%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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 75.3%

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

   if -2.79999999999999987e63 < z < -2.7999999999999999e-137

   1. Initial program 92.3%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 77.0

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

   5. Applied rewrites 77.0%

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

   if -2.7999999999999999e-137 < z < 2.40000000000000013e-54

   1. Initial program 92.6%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 86.1

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

   5. Applied rewrites 86.1%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{+63}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(t, -1, x\right)}{z}, y - a, t\right)\\ \mathbf{elif}\;z \leq -2.8 \cdot 10^{-137}:\\ \;\;\;\;\frac{t}{z - a} \cdot \left(z - y\right) + x\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-54}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(t, -1, x\right)}{z}, y - a, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 66.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{t - x}{a}, y, x\right)\\ \mathbf{if}\;a \leq -1.45 \cdot 10^{-38}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 6.5 \cdot 10^{-61}:\\ \;\;\;\;t - \frac{\left(t - x\right) \cdot y}{z}\\ \mathbf{elif}\;a \leq 4.5 \cdot 10^{+51}:\\ \;\;\;\;\frac{\left(z - y\right) \cdot t}{z - a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- t x) a) y x)))
   (if (<= a -1.45e-38)
     t_1
     (if (<= a 6.5e-61)
       (- t (/ (* (- t x) y) z))
       (if (<= a 4.5e+51) (/ (* (- z y) t) (- z a)) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((t - x) / a), y, x);
	double tmp;
	if (a <= -1.45e-38) {
		tmp = t_1;
	} else if (a <= 6.5e-61) {
		tmp = t - (((t - x) * y) / z);
	} else if (a <= 4.5e+51) {
		tmp = ((z - y) * t) / (z - a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(t - x) / a), y, x)
	tmp = 0.0
	if (a <= -1.45e-38)
		tmp = t_1;
	elseif (a <= 6.5e-61)
		tmp = Float64(t - Float64(Float64(Float64(t - x) * y) / z));
	elseif (a <= 4.5e+51)
		tmp = Float64(Float64(Float64(z - y) * t) / Float64(z - a));
	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] / a), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[a, -1.45e-38], t$95$1, If[LessEqual[a, 6.5e-61], N[(t - N[(N[(N[(t - x), $MachinePrecision] * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 4.5e+51], N[(N[(N[(z - y), $MachinePrecision] * t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.44999999999999997e-38 or 4.5e51 < a

   1. Initial program 63.6%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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 66.3

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

   5. Applied rewrites 66.3%

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

   if -1.44999999999999997e-38 < a < 6.4999999999999994e-61

   1. Initial program 71.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 71.4

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

   4. Applied rewrites 71.4%

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

   5. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 77.9

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

   7. Applied rewrites 77.9%

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

   if 6.4999999999999994e-61 < a < 4.5e51

   1. Initial program 85.8%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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. lift--.f64 N/A

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

      8. flip-- N/A

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

      9. associate-/r/ N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. difference-of-squares N/A

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

      14. lift--.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-+.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-+.f64 69.8

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

   4. Applied rewrites 69.8%

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

   5. Taylor expanded in t around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 73.9

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

   7. Applied rewrites 73.9%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.45 \cdot 10^{-38}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)\\ \mathbf{elif}\;a \leq 6.5 \cdot 10^{-61}:\\ \;\;\;\;t - \frac{\left(t - x\right) \cdot y}{z}\\ \mathbf{elif}\;a \leq 4.5 \cdot 10^{+51}:\\ \;\;\;\;\frac{\left(z - y\right) \cdot t}{z - a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{t - x}{a}, y, x\right)\\ \mathbf{if}\;a \leq -1.02 \cdot 10^{-44}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq -1.95 \cdot 10^{-305}:\\ \;\;\;\;\frac{z - y}{z} \cdot t\\ \mathbf{elif}\;a \leq 170000000000:\\ \;\;\;\;\frac{y}{z - a} \cdot \left(x - t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- t x) a) y x)))
   (if (<= a -1.02e-44)
     t_1
     (if (<= a -1.95e-305)
       (* (/ (- z y) z) t)
       (if (<= a 170000000000.0) (* (/ y (- z a)) (- x t)) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((t - x) / a), y, x);
	double tmp;
	if (a <= -1.02e-44) {
		tmp = t_1;
	} else if (a <= -1.95e-305) {
		tmp = ((z - y) / z) * t;
	} else if (a <= 170000000000.0) {
		tmp = (y / (z - a)) * (x - t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(t - x) / a), y, x)
	tmp = 0.0
	if (a <= -1.02e-44)
		tmp = t_1;
	elseif (a <= -1.95e-305)
		tmp = Float64(Float64(Float64(z - y) / z) * t);
	elseif (a <= 170000000000.0)
		tmp = Float64(Float64(y / Float64(z - a)) * Float64(x - t));
	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] / a), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[a, -1.02e-44], t$95$1, If[LessEqual[a, -1.95e-305], N[(N[(N[(z - y), $MachinePrecision] / z), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[a, 170000000000.0], N[(N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision] * N[(x - t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.0199999999999999e-44 or 1.7e11 < a

   1. Initial program 65.2%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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 64.0

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

   5. Applied rewrites 64.0%

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

   if -1.0199999999999999e-44 < a < -1.95000000000000013e-305

   1. Initial program 74.2%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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. lift--.f64 N/A

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

      8. flip-- N/A

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

      9. associate-/r/ N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. difference-of-squares N/A

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

      14. lift--.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-+.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-+.f64 54.5

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

   4. Applied rewrites 54.5%

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

   5. Taylor expanded in t around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 60.0

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

   7. Applied rewrites 60.0%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 65.8%

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

   if -1.95000000000000013e-305 < a < 1.7e11

   1. Initial program 72.5%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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 65.8

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

   5. Applied rewrites 65.8%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.02 \cdot 10^{-44}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)\\ \mathbf{elif}\;a \leq -1.95 \cdot 10^{-305}:\\ \;\;\;\;\frac{z - y}{z} \cdot t\\ \mathbf{elif}\;a \leq 170000000000:\\ \;\;\;\;\frac{y}{z - a} \cdot \left(x - t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{z - a} \cdot \left(x - t\right)\\ \mathbf{if}\;y \leq -1.85 \cdot 10^{+120}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{+91}:\\ \;\;\;\;\frac{t}{z - a} \cdot \left(z - y\right) + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ y (- z a)) (- x t))))
   (if (<= y -1.85e+120)
     t_1
     (if (<= y 6.8e+91) (+ (* (/ t (- z a)) (- z y)) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y / (z - a)) * (x - t);
	double tmp;
	if (y <= -1.85e+120) {
		tmp = t_1;
	} else if (y <= 6.8e+91) {
		tmp = ((t / (z - a)) * (z - y)) + 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 = (y / (z - a)) * (x - t)
    if (y <= (-1.85d+120)) then
        tmp = t_1
    else if (y <= 6.8d+91) then
        tmp = ((t / (z - a)) * (z - y)) + 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 = (y / (z - a)) * (x - t);
	double tmp;
	if (y <= -1.85e+120) {
		tmp = t_1;
	} else if (y <= 6.8e+91) {
		tmp = ((t / (z - a)) * (z - y)) + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y / (z - a)) * (x - t)
	tmp = 0
	if y <= -1.85e+120:
		tmp = t_1
	elif y <= 6.8e+91:
		tmp = ((t / (z - a)) * (z - y)) + x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y / Float64(z - a)) * Float64(x - t))
	tmp = 0.0
	if (y <= -1.85e+120)
		tmp = t_1;
	elseif (y <= 6.8e+91)
		tmp = Float64(Float64(Float64(t / Float64(z - a)) * Float64(z - y)) + x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y / (z - a)) * (x - t);
	tmp = 0.0;
	if (y <= -1.85e+120)
		tmp = t_1;
	elseif (y <= 6.8e+91)
		tmp = ((t / (z - a)) * (z - y)) + x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision] * N[(x - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.85e+120], t$95$1, If[LessEqual[y, 6.8e+91], N[(N[(N[(t / N[(z - a), $MachinePrecision]), $MachinePrecision] * N[(z - y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.85000000000000012e120 or 6.8000000000000002e91 < y

   1. Initial program 67.7%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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 84.0

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

   5. Applied rewrites 84.0%

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

   if -1.85000000000000012e120 < y < 6.8000000000000002e91

   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 t around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 69.8

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

   5. Applied rewrites 69.8%

      \[\leadsto x + \color{blue}{\left(y - z\right) \cdot \frac{t}{a - z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{+120}:\\ \;\;\;\;\frac{y}{z - a} \cdot \left(x - t\right)\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{+91}:\\ \;\;\;\;\frac{t}{z - a} \cdot \left(z - y\right) + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z - a} \cdot \left(x - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y - z}{a}, t - x, x\right)\\ \mathbf{if}\;a \leq -1.06 \cdot 10^{+27}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{-25}:\\ \;\;\;\;t - \frac{\left(a - y\right) \cdot \left(x - t\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- y z) a) (- t x) x)))
   (if (<= a -1.06e+27)
     t_1
     (if (<= a 1.8e-25) (- t (/ (* (- a y) (- x t)) z)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((y - z) / a), (t - x), x);
	double tmp;
	if (a <= -1.06e+27) {
		tmp = t_1;
	} else if (a <= 1.8e-25) {
		tmp = t - (((a - y) * (x - t)) / z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(y - z) / a), Float64(t - x), x)
	tmp = 0.0
	if (a <= -1.06e+27)
		tmp = t_1;
	elseif (a <= 1.8e-25)
		tmp = Float64(t - Float64(Float64(Float64(a - y) * Float64(x - t)) / z));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -1.06e+27], t$95$1, If[LessEqual[a, 1.8e-25], N[(t - N[(N[(N[(a - y), $MachinePrecision] * N[(x - t), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.05999999999999994e27 or 1.8e-25 < a

   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 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. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 70.7

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

   5. Applied rewrites 70.7%

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

   if -1.05999999999999994e27 < a < 1.8e-25

   1. Initial program 70.5%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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. lift--.f64 N/A

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

      8. flip-- N/A

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

      9. associate-/r/ N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. difference-of-squares N/A

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

      14. lift--.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-+.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-+.f64 55.1

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

   4. Applied rewrites 55.1%

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

   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. 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. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. mul-1-neg N/A

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

      5. div-sub N/A

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

      6. mul-1-neg N/A

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

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

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

      8. associate-*r/ N/A

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

      9. mul-1-neg N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 N/A

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

   7. Applied rewrites 76.6%

      \[\leadsto \color{blue}{t - \frac{\left(t - x\right) \cdot \left(y - a\right)}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.06 \cdot 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a}, t - x, x\right)\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{-25}:\\ \;\;\;\;t - \frac{\left(a - y\right) \cdot \left(x - t\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a}, t - x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y - z}{a}, t - x, x\right)\\ \mathbf{if}\;a \leq -1.3 \cdot 10^{-38}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{-25}:\\ \;\;\;\;t - \frac{\left(t - x\right) \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- y z) a) (- t x) x)))
   (if (<= a -1.3e-38) t_1 (if (<= a 1.8e-25) (- t (/ (* (- t x) y) z)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((y - z) / a), (t - x), x);
	double tmp;
	if (a <= -1.3e-38) {
		tmp = t_1;
	} else if (a <= 1.8e-25) {
		tmp = t - (((t - x) * y) / z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(y - z) / a), Float64(t - x), x)
	tmp = 0.0
	if (a <= -1.3e-38)
		tmp = t_1;
	elseif (a <= 1.8e-25)
		tmp = Float64(t - Float64(Float64(Float64(t - x) * y) / z));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -1.3e-38], t$95$1, If[LessEqual[a, 1.8e-25], N[(t - N[(N[(N[(t - x), $MachinePrecision] * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.30000000000000005e-38 or 1.8e-25 < a

   1. Initial program 67.0%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 69.1

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

   5. Applied rewrites 69.1%

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

   if -1.30000000000000005e-38 < a < 1.8e-25

   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

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 71.3

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

   4. Applied rewrites 71.3%

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

   5. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 76.5

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

   7. Applied rewrites 76.5%

      \[\leadsto \color{blue}{t - \frac{\left(t - x\right) \cdot y}{z}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 66.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{t - x}{a}, y, x\right)\\ \mathbf{if}\;a \leq -1.45 \cdot 10^{-38}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 3.5 \cdot 10^{+23}:\\ \;\;\;\;t - \frac{\left(t - x\right) \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- t x) a) y x)))
   (if (<= a -1.45e-38)
     t_1
     (if (<= a 3.5e+23) (- t (/ (* (- t x) y) z)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((t - x) / a), y, x);
	double tmp;
	if (a <= -1.45e-38) {
		tmp = t_1;
	} else if (a <= 3.5e+23) {
		tmp = t - (((t - x) * y) / z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(t - x) / a), y, x)
	tmp = 0.0
	if (a <= -1.45e-38)
		tmp = t_1;
	elseif (a <= 3.5e+23)
		tmp = Float64(t - Float64(Float64(Float64(t - x) * y) / 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] / a), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[a, -1.45e-38], t$95$1, If[LessEqual[a, 3.5e+23], N[(t - N[(N[(N[(t - x), $MachinePrecision] * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.44999999999999997e-38 or 3.5000000000000002e23 < a

   1. Initial program 64.6%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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 66.4

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

   5. Applied rewrites 66.4%

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

   if -1.44999999999999997e-38 < a < 3.5000000000000002e23

   1. Initial program 73.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 73.5

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

   4. Applied rewrites 73.5%

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

   5. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 73.7

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

   7. Applied rewrites 73.7%

      \[\leadsto \color{blue}{t - \frac{\left(t - x\right) \cdot y}{z}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 60.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{z - a} \cdot \left(z - y\right)\\ \mathbf{if}\;t \leq -1.6 \cdot 10^{-10}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{-54}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ t (- z a)) (- z y))))
   (if (<= t -1.6e-10) t_1 (if (<= t 4.8e-54) (fma (/ (- t x) a) y x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (t / (z - a)) * (z - y);
	double tmp;
	if (t <= -1.6e-10) {
		tmp = t_1;
	} else if (t <= 4.8e-54) {
		tmp = fma(((t - x) / a), y, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(t / Float64(z - a)) * Float64(z - y))
	tmp = 0.0
	if (t <= -1.6e-10)
		tmp = t_1;
	elseif (t <= 4.8e-54)
		tmp = fma(Float64(Float64(t - x) / a), y, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t / N[(z - a), $MachinePrecision]), $MachinePrecision] * N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.6e-10], t$95$1, If[LessEqual[t, 4.8e-54], N[(N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] * y + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.5999999999999999e-10 or 4.80000000000000026e-54 < t

   1. Initial program 68.4%

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

   3. Taylor expanded in t around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 72.2

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

   5. Applied rewrites 72.2%

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

   if -1.5999999999999999e-10 < t < 4.80000000000000026e-54

   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 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 52.5

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

   5. Applied rewrites 52.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.6 \cdot 10^{-10}:\\ \;\;\;\;\frac{t}{z - a} \cdot \left(z - y\right)\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{-54}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{z - a} \cdot \left(z - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{+57}:\\ \;\;\;\;\frac{z - y}{z} \cdot t\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+81}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{z - a} \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.35e+57)
   (* (/ (- z y) z) t)
   (if (<= z 5.5e+81) (fma (/ (- t x) a) y x) (* (/ z (- z a)) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.35e+57) {
		tmp = ((z - y) / z) * t;
	} else if (z <= 5.5e+81) {
		tmp = fma(((t - x) / a), y, x);
	} else {
		tmp = (z / (z - a)) * t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.35e+57)
		tmp = Float64(Float64(Float64(z - y) / z) * t);
	elseif (z <= 5.5e+81)
		tmp = fma(Float64(Float64(t - x) / a), y, x);
	else
		tmp = Float64(Float64(z / Float64(z - a)) * t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.35e+57], N[(N[(N[(z - y), $MachinePrecision] / z), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[z, 5.5e+81], N[(N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.3499999999999999e57

   1. Initial program 34.4%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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. lift--.f64 N/A

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

      8. flip-- N/A

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

      9. associate-/r/ N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. difference-of-squares N/A

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

      14. lift--.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-+.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-+.f64 19.7

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

   4. Applied rewrites 19.7%

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

   5. Taylor expanded in t around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 30.8

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

   7. Applied rewrites 30.8%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 50.1%

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

   if -1.3499999999999999e57 < z < 5.5000000000000003e81

   1. Initial program 89.1%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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 68.5

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

   5. Applied rewrites 68.5%

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

   if 5.5000000000000003e81 < z

   1. Initial program 45.1%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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. lift--.f64 N/A

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

      8. flip-- N/A

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

      9. associate-/r/ N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. difference-of-squares N/A

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

      14. lift--.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-+.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-+.f64 9.1

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

   4. Applied rewrites 9.1%

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

   5. Taylor expanded in t around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 44.1

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

   7. Applied rewrites 44.1%

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

   8. Taylor expanded in y around 0

      \[\leadsto -1 \cdot \color{blue}{\frac{t \cdot z}{a - z}} \]
   9. Step-by-step derivation

   10. Applied rewrites 52.4%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{+57}:\\ \;\;\;\;\frac{z - y}{z} \cdot t\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+81}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{z - a} \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z}{z - a} \cdot t\\ \mathbf{if}\;z \leq -9 \cdot 10^{+172}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+81}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ z (- z a)) t)))
   (if (<= z -9e+172) t_1 (if (<= z 5.5e+81) (fma (/ (- t x) a) y x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z / (z - a)) * t;
	double tmp;
	if (z <= -9e+172) {
		tmp = t_1;
	} else if (z <= 5.5e+81) {
		tmp = fma(((t - x) / a), y, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z / Float64(z - a)) * t)
	tmp = 0.0
	if (z <= -9e+172)
		tmp = t_1;
	elseif (z <= 5.5e+81)
		tmp = fma(Float64(Float64(t - x) / a), y, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[z, -9e+172], t$95$1, If[LessEqual[z, 5.5e+81], N[(N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] * y + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -9.0000000000000004e172 or 5.5000000000000003e81 < z

   1. Initial program 37.6%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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. lift--.f64 N/A

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

      8. flip-- N/A

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

      9. associate-/r/ N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. difference-of-squares N/A

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

      14. lift--.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-+.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-+.f64 5.4

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

   4. Applied rewrites 5.4%

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

   5. Taylor expanded in t around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 36.6

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

   7. Applied rewrites 36.6%

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

   8. Taylor expanded in y around 0

      \[\leadsto -1 \cdot \color{blue}{\frac{t \cdot z}{a - z}} \]
   9. Step-by-step derivation

   10. Applied rewrites 54.7%

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

   if -9.0000000000000004e172 < z < 5.5000000000000003e81

   1. Initial program 82.5%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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 63.9

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

   5. Applied rewrites 63.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{+172}:\\ \;\;\;\;\frac{z}{z - a} \cdot t\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+81}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{z - a} \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 53.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t - x\right) + x\\ \mathbf{if}\;z \leq -1.7 \cdot 10^{+173}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{+197}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- t x) x)))
   (if (<= z -1.7e+173) t_1 (if (<= z 4.8e+197) (fma (/ (- t x) a) y x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (t - x) + x;
	double tmp;
	if (z <= -1.7e+173) {
		tmp = t_1;
	} else if (z <= 4.8e+197) {
		tmp = fma(((t - x) / a), y, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(t - x) + x)
	tmp = 0.0
	if (z <= -1.7e+173)
		tmp = t_1;
	elseif (z <= 4.8e+197)
		tmp = fma(Float64(Float64(t - x) / a), y, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t - x), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -1.7e+173], t$95$1, If[LessEqual[z, 4.8e+197], N[(N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] * y + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.70000000000000011e173 or 4.7999999999999998e197 < z

   1. Initial program 28.6%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 55.4

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

   5. Applied rewrites 55.4%

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

   if -1.70000000000000011e173 < z < 4.7999999999999998e197

   1. Initial program 79.6%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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 59.5

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

   5. Applied rewrites 59.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{+173}:\\ \;\;\;\;\left(t - x\right) + x\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{+197}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t - x\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 46.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t - x\right) + x\\ \mathbf{if}\;z \leq -1.7 \cdot 10^{+173}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{+210}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- t x) x)))
   (if (<= z -1.7e+173) t_1 (if (<= z 4.5e+210) (fma y (/ t a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (t - x) + x;
	double tmp;
	if (z <= -1.7e+173) {
		tmp = t_1;
	} else if (z <= 4.5e+210) {
		tmp = fma(y, (t / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(t - x) + x)
	tmp = 0.0
	if (z <= -1.7e+173)
		tmp = t_1;
	elseif (z <= 4.5e+210)
		tmp = fma(y, Float64(t / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t - x), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -1.7e+173], t$95$1, If[LessEqual[z, 4.5e+210], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.70000000000000011e173 or 4.50000000000000004e210 < z

   1. Initial program 29.4%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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 57.4

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

   5. Applied rewrites 57.4%

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

   if -1.70000000000000011e173 < z < 4.50000000000000004e210

   1. Initial program 78.9%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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. lift--.f64 N/A

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

      8. flip-- N/A

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

      9. associate-/r/ N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. difference-of-squares N/A

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

      14. lift--.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-+.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-+.f64 63.5

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

   4. Applied rewrites 63.5%

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

   5. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{y \cdot \left(t - x\right)}{a}} \]
   6. 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. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 59.0

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

   7. Applied rewrites 59.0%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 46.8%

       \[\leadsto \mathsf{fma}\left(y, \frac{t}{\color{blue}{a}}, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{+173}:\\ \;\;\;\;\left(t - x\right) + x\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{+210}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t - x\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 27.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{z} \cdot x\\ \mathbf{if}\;y \leq -6.2 \cdot 10^{+82}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+75}:\\ \;\;\;\;\left(t - x\right) + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ y z) x)))
   (if (<= y -6.2e+82) t_1 (if (<= y 3.5e+75) (+ (- t x) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y / z) * x;
	double tmp;
	if (y <= -6.2e+82) {
		tmp = t_1;
	} else if (y <= 3.5e+75) {
		tmp = (t - x) + 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 = (y / z) * x
    if (y <= (-6.2d+82)) then
        tmp = t_1
    else if (y <= 3.5d+75) then
        tmp = (t - x) + 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 = (y / z) * x;
	double tmp;
	if (y <= -6.2e+82) {
		tmp = t_1;
	} else if (y <= 3.5e+75) {
		tmp = (t - x) + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y / z) * x
	tmp = 0
	if y <= -6.2e+82:
		tmp = t_1
	elif y <= 3.5e+75:
		tmp = (t - x) + x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y / z) * x)
	tmp = 0.0
	if (y <= -6.2e+82)
		tmp = t_1;
	elseif (y <= 3.5e+75)
		tmp = Float64(Float64(t - x) + x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y / z) * x;
	tmp = 0.0;
	if (y <= -6.2e+82)
		tmp = t_1;
	elseif (y <= 3.5e+75)
		tmp = (t - x) + x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[y, -6.2e+82], t$95$1, If[LessEqual[y, 3.5e+75], N[(N[(t - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.20000000000000065e82 or 3.4999999999999998e75 < y

   1. Initial program 69.2%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. unsub-neg N/A

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

      13. remove-double-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 50.6

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

   5. Applied rewrites 50.6%

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

   6. Taylor expanded in y around inf

      \[\leadsto -1 \cdot \color{blue}{\frac{x \cdot y}{a - z}} \]
   7. Step-by-step derivation

   8. Applied rewrites 50.4%

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

   9. Taylor expanded in a around 0

      \[\leadsto \frac{x \cdot y}{z} \]
   10. Step-by-step derivation

   11. Applied rewrites 26.8%

       \[\leadsto \frac{x \cdot y}{z} \]
   12. Step-by-step derivation

   13. Applied rewrites 36.3%

       \[\leadsto \frac{y}{z} \cdot x \]

   if -6.20000000000000065e82 < y < 3.4999999999999998e75

   1. Initial program 68.6%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 31.0

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

   5. Applied rewrites 31.0%

      \[\leadsto x + \color{blue}{\left(t - x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+82}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+75}:\\ \;\;\;\;\left(t - x\right) + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 25.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{z} \cdot y\\ \mathbf{if}\;x \leq -5.8 \cdot 10^{-98}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+131}:\\ \;\;\;\;\left(t - x\right) + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ x z) y)))
   (if (<= x -5.8e-98) t_1 (if (<= x 8.5e+131) (+ (- t x) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (x / z) * y;
	double tmp;
	if (x <= -5.8e-98) {
		tmp = t_1;
	} else if (x <= 8.5e+131) {
		tmp = (t - x) + 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 = (x / z) * y
    if (x <= (-5.8d-98)) then
        tmp = t_1
    else if (x <= 8.5d+131) then
        tmp = (t - x) + 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 = (x / z) * y;
	double tmp;
	if (x <= -5.8e-98) {
		tmp = t_1;
	} else if (x <= 8.5e+131) {
		tmp = (t - x) + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (x / z) * y
	tmp = 0
	if x <= -5.8e-98:
		tmp = t_1
	elif x <= 8.5e+131:
		tmp = (t - x) + x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(x / z) * y)
	tmp = 0.0
	if (x <= -5.8e-98)
		tmp = t_1;
	elseif (x <= 8.5e+131)
		tmp = Float64(Float64(t - x) + x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (x / z) * y;
	tmp = 0.0;
	if (x <= -5.8e-98)
		tmp = t_1;
	elseif (x <= 8.5e+131)
		tmp = (t - x) + x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[x, -5.8e-98], t$95$1, If[LessEqual[x, 8.5e+131], N[(N[(t - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.8e-98 or 8.50000000000000063e131 < x

   1. Initial program 63.5%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. unsub-neg N/A

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

      13. remove-double-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 55.9

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

   5. Applied rewrites 55.9%

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

   6. Taylor expanded in y around inf

      \[\leadsto -1 \cdot \color{blue}{\frac{x \cdot y}{a - z}} \]
   7. Step-by-step derivation

   8. Applied rewrites 42.4%

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

   9. Taylor expanded in a around 0

      \[\leadsto \frac{x \cdot y}{z} \]
   10. Step-by-step derivation

   11. Applied rewrites 27.4%

       \[\leadsto \frac{x \cdot y}{z} \]
   12. Step-by-step derivation

   13. Applied rewrites 34.9%

       \[\leadsto y \cdot \frac{x}{z} \]

   if -5.8e-98 < x < 8.50000000000000063e131

   1. Initial program 73.3%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 30.4

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

   5. Applied rewrites 30.4%

      \[\leadsto x + \color{blue}{\left(t - x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-98}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+131}:\\ \;\;\;\;\left(t - x\right) + x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 19.4% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \left(t - x\right) + x \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t, double a) {
	return (t - x) + 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 = (t - x) + x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.8%

   \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
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 21.6

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

5. Applied rewrites 21.6%

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

6. Final simplification 21.6%

   \[\leadsto \left(t - x\right) + x \]
7. Add Preprocessing

Alternative 17: 2.7% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 0.0)

C

double code(double x, double y, double z, double t, double a) {
	return 0.0;
}

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 = 0.0d0
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return 0.0;
}

Python

def code(x, y, z, t, a):
	return 0.0

Julia

function code(x, y, z, t, a)
	return 0.0
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = 0.0;
end

Wolfram

code[x_, y_, z_, t_, a_] := 0.0

Derivation

1. Initial program 68.8%

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

3. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

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

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

   6. mul-1-neg N/A

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

   7. lower-fma.f64 N/A

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

   8. mul-1-neg N/A

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

   9. sub-neg N/A

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

   10. +-commutative N/A

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

   11. distribute-neg-in N/A

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

   12. unsub-neg N/A

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

   13. remove-double-neg N/A

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

   14. lower--.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower--.f64 40.7

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

5. Applied rewrites 40.7%

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

6. Taylor expanded in z around inf

   \[\leadsto x + \color{blue}{-1 \cdot x} \]
7. Step-by-step derivation

8. Applied rewrites 2.7%

   \[\leadsto 0 \]
9. Add Preprocessing

Developer Target 1: 83.6% 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 2024263 
(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))))