Numeric.Signal:interpolate from hsignal-0.2.7.1

Percentage Accurate: 79.7% → 94.4%

Time: 13.9s

Alternatives: 16

Speedup: 0.8×

Specification

Math

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

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(y - z) * Float64(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[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 79.7% accurate, 1.0× speedup

Math

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

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(y - z) * Float64(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[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 94.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)\\ t_2 := x + \left(y - z\right) \cdot \frac{t - x}{a - z}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-261}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{z}, a - y, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- y z) (- a z)) (- t x) x))
        (t_2 (+ x (* (- y z) (/ (- t x) (- a z))))))
   (if (<= t_2 -1e-261)
     t_1
     (if (<= t_2 0.0) (fma (/ (- t x) z) (- a y) t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((y - z) / (a - z)), (t - x), x);
	double t_2 = x + ((y - z) * ((t - x) / (a - z)));
	double tmp;
	if (t_2 <= -1e-261) {
		tmp = t_1;
	} else if (t_2 <= 0.0) {
		tmp = fma(((t - x) / z), (a - y), t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(y - z) / Float64(a - z)), Float64(t - x), x)
	t_2 = Float64(x + Float64(Float64(y - z) * Float64(Float64(t - x) / Float64(a - z))))
	tmp = 0.0
	if (t_2 <= -1e-261)
		tmp = t_1;
	elseif (t_2 <= 0.0)
		tmp = fma(Float64(Float64(t - x) / z), Float64(a - y), t);
	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] / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-261], t$95$1, If[LessEqual[t$95$2, 0.0], N[(N[(N[(t - x), $MachinePrecision] / z), $MachinePrecision] * N[(a - y), $MachinePrecision] + t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < -9.99999999999999984e-262 or 0.0 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z))))

   1. Initial program 92.8%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. flip3-- N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}}}} + x \]

      7. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t \cdot t + \left(x \cdot x + t \cdot x\right)}{{t}^{3} - {x}^{3}}}} + x \]

      8. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}} + x \]

      9. flip3-- N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. --lowering--.f64 N/A

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

      14. --lowering--.f64 95.7

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

   4. Applied egg-rr 95.7%

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

   if -9.99999999999999984e-262 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 0.0

   1. Initial program 3.5%

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

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

      2. associate--l+ N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. +-commutative N/A

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

      7. associate-+r+ N/A

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

      8. associate-/l* N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt-out N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 99.7%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq -1 \cdot 10^{-261}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)\\ \mathbf{elif}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{z}, a - y, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 63.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y - z\right) \cdot \frac{t}{a - z}\\ t_2 := \mathsf{fma}\left(\frac{t - x}{z}, a, t\right)\\ \mathbf{if}\;z \leq -8 \cdot 10^{+141}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -1.15 \cdot 10^{+52}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -2.3 \cdot 10^{-151}:\\ \;\;\;\;\frac{y \cdot \left(t - x\right)}{a - z}\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{+170}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (- y z) (/ t (- a z)))) (t_2 (fma (/ (- t x) z) a t)))
   (if (<= z -8e+141)
     t_2
     (if (<= z -1.15e+52)
       t_1
       (if (<= z -2.3e-151)
         (/ (* y (- t x)) (- a z))
         (if (<= z 1.8e-19)
           (fma (/ y a) (- t x) x)
           (if (<= z 7.2e+170) t_1 t_2)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y - z) * (t / (a - z));
	double t_2 = fma(((t - x) / z), a, t);
	double tmp;
	if (z <= -8e+141) {
		tmp = t_2;
	} else if (z <= -1.15e+52) {
		tmp = t_1;
	} else if (z <= -2.3e-151) {
		tmp = (y * (t - x)) / (a - z);
	} else if (z <= 1.8e-19) {
		tmp = fma((y / a), (t - x), x);
	} else if (z <= 7.2e+170) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y - z) * Float64(t / Float64(a - z)))
	t_2 = fma(Float64(Float64(t - x) / z), a, t)
	tmp = 0.0
	if (z <= -8e+141)
		tmp = t_2;
	elseif (z <= -1.15e+52)
		tmp = t_1;
	elseif (z <= -2.3e-151)
		tmp = Float64(Float64(y * Float64(t - x)) / Float64(a - z));
	elseif (z <= 1.8e-19)
		tmp = fma(Float64(y / a), Float64(t - x), x);
	elseif (z <= 7.2e+170)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - z), $MachinePrecision] * N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t - x), $MachinePrecision] / z), $MachinePrecision] * a + t), $MachinePrecision]}, If[LessEqual[z, -8e+141], t$95$2, If[LessEqual[z, -1.15e+52], t$95$1, If[LessEqual[z, -2.3e-151], N[(N[(y * N[(t - x), $MachinePrecision]), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.8e-19], N[(N[(y / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 7.2e+170], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -8.00000000000000014e141 or 7.1999999999999999e170 < z

   1. Initial program 47.2%

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

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

      2. associate--l+ N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. +-commutative N/A

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

      7. associate-+r+ N/A

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

      8. associate-/l* N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt-out N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 87.4%

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

   6. Taylor expanded in y around 0

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

   8. Simplified 71.5%

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

   if -8.00000000000000014e141 < z < -1.15e52 or 1.8000000000000001e-19 < z < 7.1999999999999999e170

   1. Initial program 86.8%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. --lowering--.f64 52.9

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

   5. Simplified 52.9%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

         \[\leadsto t \cdot \color{blue}{\frac{1}{\frac{a - z}{y - z}}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{t \cdot 1}{\frac{a - z}{y - z}}} \]

      4. div-inv N/A

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

      5. times-frac N/A

         \[\leadsto \color{blue}{\frac{t}{a - z} \cdot \frac{1}{\frac{1}{y - z}}} \]

      6. flip-- N/A

         \[\leadsto \frac{t}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{y \cdot y - z \cdot z}{y + z}}}} \]

      7. clear-num N/A

         \[\leadsto \frac{t}{a - z} \cdot \frac{1}{\color{blue}{\frac{y + z}{y \cdot y - z \cdot z}}} \]

      8. clear-num N/A

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

      9. flip-- N/A

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

      10. *-lowering-*.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. --lowering--.f64 67.7

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

   7. Applied egg-rr 67.7%

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

   if -1.15e52 < z < -2.29999999999999996e-151

   1. Initial program 87.9%

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

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

      4. *-lowering-*.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. --lowering--.f64 62.8

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

   5. Simplified 62.8%

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

   if -2.29999999999999996e-151 < z < 1.8000000000000001e-19

   1. Initial program 93.7%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. flip3-- N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}}}} + x \]

      7. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t \cdot t + \left(x \cdot x + t \cdot x\right)}{{t}^{3} - {x}^{3}}}} + x \]

      8. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}} + x \]

      9. flip3-- N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. --lowering--.f64 N/A

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

      14. --lowering--.f64 97.8

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

   4. Applied egg-rr 97.8%

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

   5. Taylor expanded in z around 0

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

      1. /-lowering-/.f64 82.5

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

   7. Simplified 82.5%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{+141}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{z}, a, t\right)\\ \mathbf{elif}\;z \leq -1.15 \cdot 10^{+52}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{t}{a - z}\\ \mathbf{elif}\;z \leq -2.3 \cdot 10^{-151}:\\ \;\;\;\;\frac{y \cdot \left(t - x\right)}{a - z}\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{+170}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{z}, a, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 65.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y - z\right) \cdot \frac{t}{a - z}\\ t_2 := \mathsf{fma}\left(\frac{t - x}{z}, a, t\right)\\ \mathbf{if}\;z \leq -2.3 \cdot 10^{+141}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -1.4 \cdot 10^{-73}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{+170}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (- y z) (/ t (- a z)))) (t_2 (fma (/ (- t x) z) a t)))
   (if (<= z -2.3e+141)
     t_2
     (if (<= z -1.4e-73)
       t_1
       (if (<= z 1.75e-19)
         (fma (/ y a) (- t x) x)
         (if (<= z 3.9e+170) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y - z) * (t / (a - z));
	double t_2 = fma(((t - x) / z), a, t);
	double tmp;
	if (z <= -2.3e+141) {
		tmp = t_2;
	} else if (z <= -1.4e-73) {
		tmp = t_1;
	} else if (z <= 1.75e-19) {
		tmp = fma((y / a), (t - x), x);
	} else if (z <= 3.9e+170) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y - z) * Float64(t / Float64(a - z)))
	t_2 = fma(Float64(Float64(t - x) / z), a, t)
	tmp = 0.0
	if (z <= -2.3e+141)
		tmp = t_2;
	elseif (z <= -1.4e-73)
		tmp = t_1;
	elseif (z <= 1.75e-19)
		tmp = fma(Float64(y / a), Float64(t - x), x);
	elseif (z <= 3.9e+170)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - z), $MachinePrecision] * N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t - x), $MachinePrecision] / z), $MachinePrecision] * a + t), $MachinePrecision]}, If[LessEqual[z, -2.3e+141], t$95$2, If[LessEqual[z, -1.4e-73], t$95$1, If[LessEqual[z, 1.75e-19], N[(N[(y / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 3.9e+170], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.3000000000000002e141 or 3.9000000000000002e170 < z

   1. Initial program 47.2%

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

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

      2. associate--l+ N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. +-commutative N/A

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

      7. associate-+r+ N/A

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

      8. associate-/l* N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt-out N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 87.4%

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

   6. Taylor expanded in y around 0

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

   8. Simplified 71.5%

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

   if -2.3000000000000002e141 < z < -1.40000000000000006e-73 or 1.75000000000000008e-19 < z < 3.9000000000000002e170

   1. Initial program 88.3%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. --lowering--.f64 50.0

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

   5. Simplified 50.0%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

         \[\leadsto t \cdot \color{blue}{\frac{1}{\frac{a - z}{y - z}}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{t \cdot 1}{\frac{a - z}{y - z}}} \]

      4. div-inv N/A

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

      5. times-frac N/A

         \[\leadsto \color{blue}{\frac{t}{a - z} \cdot \frac{1}{\frac{1}{y - z}}} \]

      6. flip-- N/A

         \[\leadsto \frac{t}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{y \cdot y - z \cdot z}{y + z}}}} \]

      7. clear-num N/A

         \[\leadsto \frac{t}{a - z} \cdot \frac{1}{\color{blue}{\frac{y + z}{y \cdot y - z \cdot z}}} \]

      8. clear-num N/A

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

      9. flip-- N/A

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

      10. *-lowering-*.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. --lowering--.f64 60.5

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

   7. Applied egg-rr 60.5%

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

   if -1.40000000000000006e-73 < z < 1.75000000000000008e-19

   1. Initial program 91.9%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. flip3-- N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}}}} + x \]

      7. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t \cdot t + \left(x \cdot x + t \cdot x\right)}{{t}^{3} - {x}^{3}}}} + x \]

      8. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}} + x \]

      9. flip3-- N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. --lowering--.f64 N/A

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

      14. --lowering--.f64 95.5

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

   4. Applied egg-rr 95.5%

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

   5. Taylor expanded in z around 0

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

      1. /-lowering-/.f64 78.0

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

   7. Simplified 78.0%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+141}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{z}, a, t\right)\\ \mathbf{elif}\;z \leq -1.4 \cdot 10^{-73}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{t}{a - z}\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{+170}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{z}, a, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{t - x}{z}, a - y, t\right)\\ \mathbf{if}\;z \leq -1.7 \cdot 10^{+76}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 7.3 \cdot 10^{-62}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a - z}, t - x, x\right)\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+167}:\\ \;\;\;\;x + \left(y - z\right) \cdot \frac{t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- t x) z) (- a y) t)))
   (if (<= z -1.7e+76)
     t_1
     (if (<= z 7.3e-62)
       (fma (/ y (- a z)) (- t x) x)
       (if (<= z 7e+167) (+ x (* (- y z) (/ t (- a z)))) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((t - x) / z), (a - y), t);
	double tmp;
	if (z <= -1.7e+76) {
		tmp = t_1;
	} else if (z <= 7.3e-62) {
		tmp = fma((y / (a - z)), (t - x), x);
	} else if (z <= 7e+167) {
		tmp = x + ((y - z) * (t / (a - z)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(t - x) / z), Float64(a - y), t)
	tmp = 0.0
	if (z <= -1.7e+76)
		tmp = t_1;
	elseif (z <= 7.3e-62)
		tmp = fma(Float64(y / Float64(a - z)), Float64(t - x), x);
	elseif (z <= 7e+167)
		tmp = Float64(x + Float64(Float64(y - z) * Float64(t / Float64(a - z))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(t - x), $MachinePrecision] / z), $MachinePrecision] * N[(a - y), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -1.7e+76], t$95$1, If[LessEqual[z, 7.3e-62], N[(N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 7e+167], N[(x + N[(N[(y - z), $MachinePrecision] * N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.6999999999999999e76 or 6.99999999999999975e167 < z

   1. Initial program 54.8%

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

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

      2. associate--l+ N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. +-commutative N/A

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

      7. associate-+r+ N/A

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

      8. associate-/l* N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt-out N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 85.7%

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

   if -1.6999999999999999e76 < z < 7.2999999999999998e-62

   1. Initial program 91.6%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. flip3-- N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}}}} + x \]

      7. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t \cdot t + \left(x \cdot x + t \cdot x\right)}{{t}^{3} - {x}^{3}}}} + x \]

      8. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}} + x \]

      9. flip3-- N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. --lowering--.f64 N/A

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

      14. --lowering--.f64 93.7

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

   4. Applied egg-rr 93.7%

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

   5. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 82.8

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

   7. Simplified 82.8%

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

   if 7.2999999999999998e-62 < z < 6.99999999999999975e167

   1. Initial program 86.2%

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

   3. Taylor expanded in t around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 78.7

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

   5. Simplified 78.7%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{+76}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{z}, a - y, t\right)\\ \mathbf{elif}\;z \leq 7.3 \cdot 10^{-62}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a - z}, t - x, x\right)\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+167}:\\ \;\;\;\;x + \left(y - z\right) \cdot \frac{t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{z}, a - y, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{a - y}{z}, t - x, t\right)\\ \mathbf{if}\;z \leq -6 \cdot 10^{+72}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.15 \cdot 10^{-61}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a - z}, t - x, x\right)\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{+159}:\\ \;\;\;\;x + \left(y - z\right) \cdot \frac{t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- a y) z) (- t x) t)))
   (if (<= z -6e+72)
     t_1
     (if (<= z 2.15e-61)
       (fma (/ y (- a z)) (- t x) x)
       (if (<= z 2.9e+159) (+ x (* (- y z) (/ t (- a z)))) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((a - y) / z), (t - x), t);
	double tmp;
	if (z <= -6e+72) {
		tmp = t_1;
	} else if (z <= 2.15e-61) {
		tmp = fma((y / (a - z)), (t - x), x);
	} else if (z <= 2.9e+159) {
		tmp = x + ((y - z) * (t / (a - z)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(a - y) / z), Float64(t - x), t)
	tmp = 0.0
	if (z <= -6e+72)
		tmp = t_1;
	elseif (z <= 2.15e-61)
		tmp = fma(Float64(y / Float64(a - z)), Float64(t - x), x);
	elseif (z <= 2.9e+159)
		tmp = Float64(x + Float64(Float64(y - z) * Float64(t / Float64(a - z))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(a - y), $MachinePrecision] / z), $MachinePrecision] * N[(t - x), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -6e+72], t$95$1, If[LessEqual[z, 2.15e-61], N[(N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 2.9e+159], N[(x + N[(N[(y - z), $MachinePrecision] * N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.00000000000000006e72 or 2.90000000000000014e159 < z

   1. Initial program 54.8%

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

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

      2. associate--l+ N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. +-commutative N/A

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

      7. associate-+r+ N/A

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

      8. associate-/l* N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt-out N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 85.7%

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

   6. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. associate-/l* N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. associate-/l* N/A

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

      9. associate-*r* N/A

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

      10. mul-1-neg N/A

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

      11. associate-/l* N/A

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

      12. distribute-rgt-out N/A

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

      13. +-commutative N/A

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

      14. sub-neg N/A

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

      15. accelerator-lowering-fma.f64 N/A

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

   8. Simplified 84.3%

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

   if -6.00000000000000006e72 < z < 2.1500000000000002e-61

   1. Initial program 91.6%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. flip3-- N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}}}} + x \]

      7. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t \cdot t + \left(x \cdot x + t \cdot x\right)}{{t}^{3} - {x}^{3}}}} + x \]

      8. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}} + x \]

      9. flip3-- N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. --lowering--.f64 N/A

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

      14. --lowering--.f64 93.7

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

   4. Applied egg-rr 93.7%

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

   5. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 82.8

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

   7. Simplified 82.8%

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

   if 2.1500000000000002e-61 < z < 2.90000000000000014e159

   1. Initial program 86.2%

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

   3. Taylor expanded in t around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 78.7

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

   5. Simplified 78.7%

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

Alternative 6: 80.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{a - y}{z}, t - x, t\right)\\ \mathbf{if}\;z \leq -1.7 \cdot 10^{+73}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 190:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a - z}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- a y) z) (- t x) t)))
   (if (<= z -1.7e+73)
     t_1
     (if (<= z 190.0) (fma (/ y (- a z)) (- t x) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((a - y) / z), (t - x), t);
	double tmp;
	if (z <= -1.7e+73) {
		tmp = t_1;
	} else if (z <= 190.0) {
		tmp = fma((y / (a - z)), (t - x), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(a - y) / z), Float64(t - x), t)
	tmp = 0.0
	if (z <= -1.7e+73)
		tmp = t_1;
	elseif (z <= 190.0)
		tmp = fma(Float64(y / Float64(a - z)), 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[(a - y), $MachinePrecision] / z), $MachinePrecision] * N[(t - x), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -1.7e+73], t$95$1, If[LessEqual[z, 190.0], N[(N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.7000000000000001e73 or 190 < z

   1. Initial program 62.4%

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

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

      2. associate--l+ N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. +-commutative N/A

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

      7. associate-+r+ N/A

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

      8. associate-/l* N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt-out N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 81.1%

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

   6. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. associate-/l* N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. associate-/l* N/A

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

      9. associate-*r* N/A

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

      10. mul-1-neg N/A

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

      11. associate-/l* N/A

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

      12. distribute-rgt-out N/A

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

      13. +-commutative N/A

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

      14. sub-neg N/A

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

      15. accelerator-lowering-fma.f64 N/A

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

   8. Simplified 80.1%

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

   if -1.7000000000000001e73 < z < 190

   1. Initial program 92.0%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. flip3-- N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}}}} + x \]

      7. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t \cdot t + \left(x \cdot x + t \cdot x\right)}{{t}^{3} - {x}^{3}}}} + x \]

      8. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}} + x \]

      9. flip3-- N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. --lowering--.f64 N/A

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

      14. --lowering--.f64 93.2

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

   4. Applied egg-rr 93.2%

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

   5. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 80.4

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

   7. Simplified 80.4%

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

Alternative 7: 76.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{t - x}{z}, -y, t\right)\\ \mathbf{if}\;z \leq -1.9 \cdot 10^{+72}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a - z}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- t x) z) (- y) t)))
   (if (<= z -1.9e+72)
     t_1
     (if (<= z 4.4e-8) (fma (/ y (- a z)) (- t x) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((t - x) / z), -y, t);
	double tmp;
	if (z <= -1.9e+72) {
		tmp = t_1;
	} else if (z <= 4.4e-8) {
		tmp = fma((y / (a - z)), (t - x), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(t - x) / z), Float64(-y), t)
	tmp = 0.0
	if (z <= -1.9e+72)
		tmp = t_1;
	elseif (z <= 4.4e-8)
		tmp = fma(Float64(y / Float64(a - z)), 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 - x), $MachinePrecision] / z), $MachinePrecision] * (-y) + t), $MachinePrecision]}, If[LessEqual[z, -1.9e+72], t$95$1, If[LessEqual[z, 4.4e-8], N[(N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.90000000000000003e72 or 4.3999999999999997e-8 < z

   1. Initial program 64.7%

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

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

      2. associate--l+ N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. +-commutative N/A

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

      7. associate-+r+ N/A

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

      8. associate-/l* N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt-out N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 79.7%

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

   6. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 68.6

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

   8. Simplified 68.6%

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

   if -1.90000000000000003e72 < z < 4.3999999999999997e-8

   1. Initial program 91.6%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. flip3-- N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}}}} + x \]

      7. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t \cdot t + \left(x \cdot x + t \cdot x\right)}{{t}^{3} - {x}^{3}}}} + x \]

      8. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}} + x \]

      9. flip3-- N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. --lowering--.f64 N/A

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

      14. --lowering--.f64 93.5

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

   4. Applied egg-rr 93.5%

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

   5. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 82.1

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

   7. Simplified 82.1%

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

Alternative 8: 72.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)\\ \mathbf{if}\;a \leq -4400:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 3.2 \cdot 10^{-52}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{z}, -y, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- y z) (/ (- t x) a) x)))
   (if (<= a -4400.0)
     t_1
     (if (<= a 3.2e-52) (fma (/ (- t x) z) (- y) t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((y - z), ((t - x) / a), x);
	double tmp;
	if (a <= -4400.0) {
		tmp = t_1;
	} else if (a <= 3.2e-52) {
		tmp = fma(((t - x) / z), -y, t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(y - z), Float64(Float64(t - x) / a), x)
	tmp = 0.0
	if (a <= -4400.0)
		tmp = t_1;
	elseif (a <= 3.2e-52)
		tmp = fma(Float64(Float64(t - x) / z), Float64(-y), t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -4400.0], t$95$1, If[LessEqual[a, 3.2e-52], N[(N[(N[(t - x), $MachinePrecision] / z), $MachinePrecision] * (-y) + t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -4400 or 3.2000000000000001e-52 < a

   1. Initial program 86.5%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. --lowering--.f64 72.4

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

   5. Simplified 72.4%

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

   if -4400 < a < 3.2000000000000001e-52

   1. Initial program 72.6%

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

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

      2. associate--l+ N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. +-commutative N/A

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

      7. associate-+r+ N/A

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

      8. associate-/l* N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt-out N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 81.6%

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

   6. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 77.6

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

   8. Simplified 77.6%

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

Alternative 9: 68.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t, \frac{y - z}{a}, x\right)\\ \mathbf{if}\;a \leq -195:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 3.2 \cdot 10^{-52}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{z}, -y, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma t (/ (- y z) a) x)))
   (if (<= a -195.0) t_1 (if (<= a 3.2e-52) (fma (/ (- t x) z) (- y) t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(t, ((y - z) / a), x);
	double tmp;
	if (a <= -195.0) {
		tmp = t_1;
	} else if (a <= 3.2e-52) {
		tmp = fma(((t - x) / z), -y, t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(t, Float64(Float64(y - z) / a), x)
	tmp = 0.0
	if (a <= -195.0)
		tmp = t_1;
	elseif (a <= 3.2e-52)
		tmp = fma(Float64(Float64(t - x) / z), Float64(-y), t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t * N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -195.0], t$95$1, If[LessEqual[a, 3.2e-52], N[(N[(N[(t - x), $MachinePrecision] / z), $MachinePrecision] * (-y) + t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -195 or 3.2000000000000001e-52 < a

   1. Initial program 86.5%

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

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. --lowering--.f64 60.4

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

   5. Simplified 60.4%

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

   6. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. --lowering--.f64 65.8

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

   8. Simplified 65.8%

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

   if -195 < a < 3.2000000000000001e-52

   1. Initial program 72.6%

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

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

      2. associate--l+ N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. +-commutative N/A

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

      7. associate-+r+ N/A

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

      8. associate-/l* N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt-out N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 81.6%

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

   6. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 77.6

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

   8. Simplified 77.6%

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

Alternative 10: 63.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{t - x}{z}, a, t\right)\\ \mathbf{if}\;z \leq -6.4 \cdot 10^{+85}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+64}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- t x) z) a t)))
   (if (<= z -6.4e+85) t_1 (if (<= z 1.1e+64) (fma (/ y a) (- t x) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((t - x) / z), a, t);
	double tmp;
	if (z <= -6.4e+85) {
		tmp = t_1;
	} else if (z <= 1.1e+64) {
		tmp = fma((y / a), (t - x), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(t - x) / z), a, t)
	tmp = 0.0
	if (z <= -6.4e+85)
		tmp = t_1;
	elseif (z <= 1.1e+64)
		tmp = fma(Float64(y / 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 - x), $MachinePrecision] / z), $MachinePrecision] * a + t), $MachinePrecision]}, If[LessEqual[z, -6.4e+85], t$95$1, If[LessEqual[z, 1.1e+64], N[(N[(y / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.40000000000000037e85 or 1.10000000000000001e64 < z

   1. Initial program 60.9%

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

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

      2. associate--l+ N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. +-commutative N/A

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

      7. associate-+r+ N/A

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

      8. associate-/l* N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt-out N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 82.3%

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

   6. Taylor expanded in y around 0

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

   8. Simplified 62.4%

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

   if -6.40000000000000037e85 < z < 1.10000000000000001e64

   1. Initial program 91.2%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. flip3-- N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}}}} + x \]

      7. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t \cdot t + \left(x \cdot x + t \cdot x\right)}{{t}^{3} - {x}^{3}}}} + x \]

      8. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}} + x \]

      9. flip3-- N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. --lowering--.f64 N/A

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

      14. --lowering--.f64 92.4

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

   4. Applied egg-rr 92.4%

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

   5. Taylor expanded in z around 0

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

      1. /-lowering-/.f64 64.9

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

   7. Simplified 64.9%

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

Alternative 11: 59.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t, \frac{y - z}{a}, x\right)\\ \mathbf{if}\;a \leq -64:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 7 \cdot 10^{-36}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{a - y}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma t (/ (- y z) a) x)))
   (if (<= a -64.0) t_1 (if (<= a 7e-36) (fma t (/ (- a y) z) t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(t, ((y - z) / a), x);
	double tmp;
	if (a <= -64.0) {
		tmp = t_1;
	} else if (a <= 7e-36) {
		tmp = fma(t, ((a - y) / z), t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(t, Float64(Float64(y - z) / a), x)
	tmp = 0.0
	if (a <= -64.0)
		tmp = t_1;
	elseif (a <= 7e-36)
		tmp = fma(t, Float64(Float64(a - y) / z), t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t * N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -64.0], t$95$1, If[LessEqual[a, 7e-36], N[(t * N[(N[(a - y), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -64 or 6.9999999999999999e-36 < a

   1. Initial program 86.4%

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

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. --lowering--.f64 60.9

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

   5. Simplified 60.9%

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

   6. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. --lowering--.f64 66.3

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

   8. Simplified 66.3%

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

   if -64 < a < 6.9999999999999999e-36

   1. Initial program 72.8%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. --lowering--.f64 51.9

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

   5. Simplified 51.9%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

         \[\leadsto t \cdot \color{blue}{\frac{1}{\frac{a - z}{y - z}}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{t \cdot 1}{\frac{a - z}{y - z}}} \]

      4. div-inv N/A

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

      5. times-frac N/A

         \[\leadsto \color{blue}{\frac{t}{a - z} \cdot \frac{1}{\frac{1}{y - z}}} \]

      6. flip-- N/A

         \[\leadsto \frac{t}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{y \cdot y - z \cdot z}{y + z}}}} \]

      7. clear-num N/A

         \[\leadsto \frac{t}{a - z} \cdot \frac{1}{\color{blue}{\frac{y + z}{y \cdot y - z \cdot z}}} \]

      8. clear-num N/A

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

      9. flip-- N/A

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

      10. *-lowering-*.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. --lowering--.f64 53.2

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

   7. Applied egg-rr 53.2%

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

   8. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{t \cdot y - a \cdot t}{z}} \]

      4. associate-*r/ N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. distribute-neg-out N/A

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

      8. *-commutative N/A

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

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

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

      10. mul-1-neg N/A

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

      11. remove-double-neg N/A

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

      12. distribute-rgt-in N/A

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

      13. +-commutative N/A

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

      14. mul-1-neg N/A

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

      15. sub-neg N/A

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

      16. +-commutative N/A

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

      17. associate-/l* N/A

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

   10. Simplified 55.9%

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

Alternative 12: 56.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \mathbf{if}\;a \leq -5500:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 6.9 \cdot 10^{-40}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{a - y}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma t (/ y a) x)))
   (if (<= a -5500.0) t_1 (if (<= a 6.9e-40) (fma t (/ (- a y) z) t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(t, (y / a), x);
	double tmp;
	if (a <= -5500.0) {
		tmp = t_1;
	} else if (a <= 6.9e-40) {
		tmp = fma(t, ((a - y) / z), t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(t, Float64(y / a), x)
	tmp = 0.0
	if (a <= -5500.0)
		tmp = t_1;
	elseif (a <= 6.9e-40)
		tmp = fma(t, Float64(Float64(a - y) / z), t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t * N[(y / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -5500.0], t$95$1, If[LessEqual[a, 6.9e-40], N[(t * N[(N[(a - y), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -5500 or 6.8999999999999996e-40 < a

   1. Initial program 86.4%

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

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. --lowering--.f64 60.9

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

   5. Simplified 60.9%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. /-lowering-/.f64 56.7

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

   8. Simplified 56.7%

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

   if -5500 < a < 6.8999999999999996e-40

   1. Initial program 72.8%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. --lowering--.f64 51.9

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

   5. Simplified 51.9%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

         \[\leadsto t \cdot \color{blue}{\frac{1}{\frac{a - z}{y - z}}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{t \cdot 1}{\frac{a - z}{y - z}}} \]

      4. div-inv N/A

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

      5. times-frac N/A

         \[\leadsto \color{blue}{\frac{t}{a - z} \cdot \frac{1}{\frac{1}{y - z}}} \]

      6. flip-- N/A

         \[\leadsto \frac{t}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{y \cdot y - z \cdot z}{y + z}}}} \]

      7. clear-num N/A

         \[\leadsto \frac{t}{a - z} \cdot \frac{1}{\color{blue}{\frac{y + z}{y \cdot y - z \cdot z}}} \]

      8. clear-num N/A

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

      9. flip-- N/A

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

      10. *-lowering-*.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. --lowering--.f64 53.2

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

   7. Applied egg-rr 53.2%

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

   8. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{t \cdot y - a \cdot t}{z}} \]

      4. associate-*r/ N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. distribute-neg-out N/A

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

      8. *-commutative N/A

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

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

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

      10. mul-1-neg N/A

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

      11. remove-double-neg N/A

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

      12. distribute-rgt-in N/A

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

      13. +-commutative N/A

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

      14. mul-1-neg N/A

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

      15. sub-neg N/A

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

      16. +-commutative N/A

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

      17. associate-/l* N/A

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

   10. Simplified 55.9%

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

Alternative 13: 53.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+76}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{+64}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -4e+76) t (if (<= z 1.02e+64) (fma t (/ y a) x) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4e+76) {
		tmp = t;
	} else if (z <= 1.02e+64) {
		tmp = fma(t, (y / a), x);
	} else {
		tmp = t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -4e+76)
		tmp = t;
	elseif (z <= 1.02e+64)
		tmp = fma(t, Float64(y / a), x);
	else
		tmp = t;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -4e+76], t, If[LessEqual[z, 1.02e+64], N[(t * N[(y / a), $MachinePrecision] + x), $MachinePrecision], t]]

Derivation

1. Split input into 2 regimes

2. if z < -4.0000000000000002e76 or 1.01999999999999996e64 < z

   1. Initial program 60.9%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{t} \]
   4. Step-by-step derivation

   5. Simplified 50.1%

      \[\leadsto \color{blue}{t} \]

   if -4.0000000000000002e76 < z < 1.01999999999999996e64

   1. Initial program 91.2%

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

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. --lowering--.f64 68.6

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

   5. Simplified 68.6%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. /-lowering-/.f64 54.5

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

   8. Simplified 54.5%

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

Alternative 14: 37.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.7 \cdot 10^{+51}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq -1.3 \cdot 10^{-153}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;z \leq 1.98 \cdot 10^{-19}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -4.7e+51)
   t
   (if (<= z -1.3e-153) (/ (* x y) z) (if (<= z 1.98e-19) x t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4.7e+51) {
		tmp = t;
	} else if (z <= -1.3e-153) {
		tmp = (x * y) / z;
	} else if (z <= 1.98e-19) {
		tmp = x;
	} else {
		tmp = t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-4.7d+51)) then
        tmp = t
    else if (z <= (-1.3d-153)) then
        tmp = (x * y) / z
    else if (z <= 1.98d-19) then
        tmp = x
    else
        tmp = t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4.7e+51) {
		tmp = t;
	} else if (z <= -1.3e-153) {
		tmp = (x * y) / z;
	} else if (z <= 1.98e-19) {
		tmp = x;
	} else {
		tmp = t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -4.7e+51:
		tmp = t
	elif z <= -1.3e-153:
		tmp = (x * y) / z
	elif z <= 1.98e-19:
		tmp = x
	else:
		tmp = t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -4.7e+51)
		tmp = t;
	elseif (z <= -1.3e-153)
		tmp = Float64(Float64(x * y) / z);
	elseif (z <= 1.98e-19)
		tmp = x;
	else
		tmp = t;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -4.7e+51)
		tmp = t;
	elseif (z <= -1.3e-153)
		tmp = (x * y) / z;
	elseif (z <= 1.98e-19)
		tmp = x;
	else
		tmp = t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -4.7e+51], t, If[LessEqual[z, -1.3e-153], N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 1.98e-19], x, t]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.7000000000000002e51 or 1.98000000000000001e-19 < z

   1. Initial program 66.8%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{t} \]
   4. Step-by-step derivation

   5. Simplified 45.0%

      \[\leadsto \color{blue}{t} \]

   if -4.7000000000000002e51 < z < -1.3000000000000001e-153

   1. Initial program 88.3%

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

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

      2. associate--l+ N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. +-commutative N/A

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

      7. associate-+r+ N/A

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

      8. associate-/l* N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt-out N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 57.0%

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

   6. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. neg-lowering-neg.f64 35.6

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

   8. Simplified 35.6%

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

   9. Taylor expanded in a around 0

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

       1. /-lowering-/.f64 N/A

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 31.2

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

   11. Simplified 31.2%

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

   if -1.3000000000000001e-153 < z < 1.98000000000000001e-19

   1. Initial program 93.6%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 40.7%

      \[\leadsto \color{blue}{x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 41.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.7 \cdot 10^{+51}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq -1.3 \cdot 10^{-153}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;z \leq 1.98 \cdot 10^{-19}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 38.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{+72}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{-19}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -7.2e+72) t (if (<= z 1.75e-19) x t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -7.2e+72) {
		tmp = t;
	} else if (z <= 1.75e-19) {
		tmp = x;
	} else {
		tmp = t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-7.2d+72)) then
        tmp = t
    else if (z <= 1.75d-19) then
        tmp = x
    else
        tmp = t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -7.2e+72) {
		tmp = t;
	} else if (z <= 1.75e-19) {
		tmp = x;
	} else {
		tmp = t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -7.2e+72:
		tmp = t
	elif z <= 1.75e-19:
		tmp = x
	else:
		tmp = t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -7.2e+72)
		tmp = t;
	elseif (z <= 1.75e-19)
		tmp = x;
	else
		tmp = t;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -7.2e+72)
		tmp = t;
	elseif (z <= 1.75e-19)
		tmp = x;
	else
		tmp = t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -7.2e+72], t, If[LessEqual[z, 1.75e-19], x, t]]

Derivation

1. Split input into 2 regimes

2. if z < -7.20000000000000069e72 or 1.75000000000000008e-19 < z

   1. Initial program 66.2%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{t} \]
   4. Step-by-step derivation

   5. Simplified 46.7%

      \[\leadsto \color{blue}{t} \]

   if -7.20000000000000069e72 < z < 1.75000000000000008e-19

   1. Initial program 91.3%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 32.3%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 25.8% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.5%

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

3. Taylor expanded in z around inf

   \[\leadsto \color{blue}{t} \]
4. Step-by-step derivation

5. Simplified 25.6%

   \[\leadsto \color{blue}{t} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024205 
(FPCore (x y z t a)
  :name "Numeric.Signal:interpolate   from hsignal-0.2.7.1"
  :precision binary64
  (+ x (* (- y z) (/ (- t x) (- a z)))))