Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, C

Percentage Accurate: 94.5% → 96.0%

Time: 10.6s

Alternatives: 12

Speedup: 0.7×

Specification

Math

\[\begin{array}{l} \\ x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* x (- (/ y z) (/ t (- 1.0 z)))))

C

double code(double x, double y, double z, double t) {
	return x * ((y / z) - (t / (1.0 - z)));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x * ((y / z) - (t / (1.0d0 - z)))
end function

Java

public static double code(double x, double y, double z, double t) {
	return x * ((y / z) - (t / (1.0 - z)));
}

Python

def code(x, y, z, t):
	return x * ((y / z) - (t / (1.0 - z)))

Julia

function code(x, y, z, t)
	return Float64(x * Float64(Float64(y / z) - Float64(t / Float64(1.0 - z))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x * ((y / z) - (t / (1.0 - z)));
end

Wolfram

code[x_, y_, z_, t_] := N[(x * N[(N[(y / z), $MachinePrecision] - N[(t / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 94.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* x (- (/ y z) (/ t (- 1.0 z)))))

C

double code(double x, double y, double z, double t) {
	return x * ((y / z) - (t / (1.0 - z)));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x * ((y / z) - (t / (1.0d0 - z)))
end function

Java

public static double code(double x, double y, double z, double t) {
	return x * ((y / z) - (t / (1.0 - z)));
}

Python

def code(x, y, z, t):
	return x * ((y / z) - (t / (1.0 - z)))

Julia

function code(x, y, z, t)
	return Float64(x * Float64(Float64(y / z) - Float64(t / Float64(1.0 - z))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x * ((y / z) - (t / (1.0 - z)));
end

Wolfram

code[x_, y_, z_, t_] := N[(x * N[(N[(y / z), $MachinePrecision] - N[(t / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 96.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5 \cdot 10^{-121}:\\ \;\;\;\;\mathsf{fma}\left(x\_m \cdot t, \frac{1}{z + -1}, \frac{x\_m \cdot y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(\frac{y}{z} + \frac{1}{\frac{z + -1}{t}}\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= x_m 5e-121)
    (fma (* x_m t) (/ 1.0 (+ z -1.0)) (/ (* x_m y) z))
    (* x_m (+ (/ y z) (/ 1.0 (/ (+ z -1.0) t)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (x_m <= 5e-121) {
		tmp = fma((x_m * t), (1.0 / (z + -1.0)), ((x_m * y) / z));
	} else {
		tmp = x_m * ((y / z) + (1.0 / ((z + -1.0) / t)));
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (x_m <= 5e-121)
		tmp = fma(Float64(x_m * t), Float64(1.0 / Float64(z + -1.0)), Float64(Float64(x_m * y) / z));
	else
		tmp = Float64(x_m * Float64(Float64(y / z) + Float64(1.0 / Float64(Float64(z + -1.0) / t))));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[x$95$m, 5e-121], N[(N[(x$95$m * t), $MachinePrecision] * N[(1.0 / N[(z + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x$95$m * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(N[(y / z), $MachinePrecision] + N[(1.0 / N[(N[(z + -1.0), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 4.99999999999999989e-121

   1. Initial program 91.8%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 91.8

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

   4. Applied rewrites 91.8%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. distribute-lft-in N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. distribute-frac-neg N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. distribute-neg-in N/A

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

      13. lift-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. frac-2neg N/A

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

      18. div-inv N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lift-+.f64 N/A

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

   6. Applied rewrites 93.5%

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

   if 4.99999999999999989e-121 < x

   1. Initial program 97.7%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 97.7

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

   4. Applied rewrites 97.7%

      \[\leadsto x \cdot \left(\frac{y}{z} - \color{blue}{\frac{1}{\frac{1 - z}{t}}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-121}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot t, \frac{1}{z + -1}, \frac{x \cdot y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} + \frac{1}{\frac{z + -1}{t}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \frac{y}{z} + \frac{t}{z + -1}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{x\_m \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (+ (/ y z) (/ t (+ z -1.0)))))
   (* x_s (if (<= t_1 (- INFINITY)) (/ (* x_m y) z) (* x_m t_1)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (y / z) + (t / (z + -1.0));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (x_m * y) / z;
	} else {
		tmp = x_m * t_1;
	}
	return x_s * tmp;
}

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (y / z) + (t / (z + -1.0));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = (x_m * y) / z;
	} else {
		tmp = x_m * t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	t_1 = (y / z) + (t / (z + -1.0))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = (x_m * y) / z
	else:
		tmp = x_m * t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(Float64(y / z) + Float64(t / Float64(z + -1.0)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(x_m * y) / z);
	else
		tmp = Float64(x_m * t_1);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = (y / z) + (t / (z + -1.0));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = (x_m * y) / z;
	else
		tmp = x_m * t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y / z), $MachinePrecision] + N[(t / N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(N[(x$95$m * y), $MachinePrecision] / z), $MachinePrecision], N[(x$95$m * t$95$1), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z))) < -inf.0

   1. Initial program 56.8%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

   if -inf.0 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z)))

   1. Initial program 96.2%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} + \frac{t}{z + -1} \leq -\infty:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} + \frac{t}{z + -1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5 \cdot 10^{-121}:\\ \;\;\;\;\mathsf{fma}\left(x\_m \cdot t, \frac{1}{z + -1}, \frac{x\_m \cdot y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(\frac{y}{z} + \frac{t}{z + -1}\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= x_m 5e-121)
    (fma (* x_m t) (/ 1.0 (+ z -1.0)) (/ (* x_m y) z))
    (* x_m (+ (/ y z) (/ t (+ z -1.0)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (x_m <= 5e-121) {
		tmp = fma((x_m * t), (1.0 / (z + -1.0)), ((x_m * y) / z));
	} else {
		tmp = x_m * ((y / z) + (t / (z + -1.0)));
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (x_m <= 5e-121)
		tmp = fma(Float64(x_m * t), Float64(1.0 / Float64(z + -1.0)), Float64(Float64(x_m * y) / z));
	else
		tmp = Float64(x_m * Float64(Float64(y / z) + Float64(t / Float64(z + -1.0))));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[x$95$m, 5e-121], N[(N[(x$95$m * t), $MachinePrecision] * N[(1.0 / N[(z + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x$95$m * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(N[(y / z), $MachinePrecision] + N[(t / N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 4.99999999999999989e-121

   1. Initial program 91.8%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 91.8

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

   4. Applied rewrites 91.8%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. distribute-lft-in N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. distribute-frac-neg N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. distribute-neg-in N/A

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

      13. lift-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. frac-2neg N/A

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

      18. div-inv N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lift-+.f64 N/A

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

   6. Applied rewrites 93.5%

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

   if 4.99999999999999989e-121 < x

   1. Initial program 97.7%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-121}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot t, \frac{1}{z + -1}, \frac{x \cdot y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} + \frac{t}{z + -1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -0.75:\\ \;\;\;\;\frac{x\_m}{z} \cdot \left(t + y\right)\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x\_m \cdot \left(\frac{y}{z} - \mathsf{fma}\left(z, t, t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot \left(t + y\right)}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z -0.75)
    (* (/ x_m z) (+ t y))
    (if (<= z 1.0) (* x_m (- (/ y z) (fma z t t))) (/ (* x_m (+ t y)) z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -0.75) {
		tmp = (x_m / z) * (t + y);
	} else if (z <= 1.0) {
		tmp = x_m * ((y / z) - fma(z, t, t));
	} else {
		tmp = (x_m * (t + y)) / z;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= -0.75)
		tmp = Float64(Float64(x_m / z) * Float64(t + y));
	elseif (z <= 1.0)
		tmp = Float64(x_m * Float64(Float64(y / z) - fma(z, t, t)));
	else
		tmp = Float64(Float64(x_m * Float64(t + y)) / z);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, -0.75], N[(N[(x$95$m / z), $MachinePrecision] * N[(t + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.0], N[(x$95$m * N[(N[(y / z), $MachinePrecision] - N[(z * t + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * N[(t + y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -0.75

   1. Initial program 96.3%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. remove-double-neg N/A

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

      3. neg-mul-1 N/A

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

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

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

      5. neg-mul-1 N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

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

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

      9. *-commutative N/A

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

      10. mul-1-neg N/A

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

      11. lower-/.f64 N/A

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

   5. Applied rewrites 85.2%

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

   7. Applied rewrites 87.9%

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

   if -0.75 < z < 1

   1. Initial program 92.4%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 92.3

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

   5. Applied rewrites 92.3%

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

   if 1 < z

   1. Initial program 94.9%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. remove-double-neg N/A

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

      3. neg-mul-1 N/A

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

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

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

      5. neg-mul-1 N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

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

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

      9. *-commutative N/A

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

      10. mul-1-neg N/A

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

      11. lower-/.f64 N/A

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

   5. Applied rewrites 89.3%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.75:\\ \;\;\;\;\frac{x}{z} \cdot \left(t + y\right)\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \mathsf{fma}\left(z, t, t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(t + y\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -0.85:\\ \;\;\;\;\frac{x\_m}{z} \cdot \left(t + y\right)\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x\_m \cdot \left(\frac{y}{z} + \left(-t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot \left(t + y\right)}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z -0.85)
    (* (/ x_m z) (+ t y))
    (if (<= z 1.0) (* x_m (+ (/ y z) (- t))) (/ (* x_m (+ t y)) z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -0.85) {
		tmp = (x_m / z) * (t + y);
	} else if (z <= 1.0) {
		tmp = x_m * ((y / z) + -t);
	} else {
		tmp = (x_m * (t + y)) / z;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-0.85d0)) then
        tmp = (x_m / z) * (t + y)
    else if (z <= 1.0d0) then
        tmp = x_m * ((y / z) + -t)
    else
        tmp = (x_m * (t + y)) / z
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -0.85) {
		tmp = (x_m / z) * (t + y);
	} else if (z <= 1.0) {
		tmp = x_m * ((y / z) + -t);
	} else {
		tmp = (x_m * (t + y)) / z;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if z <= -0.85:
		tmp = (x_m / z) * (t + y)
	elif z <= 1.0:
		tmp = x_m * ((y / z) + -t)
	else:
		tmp = (x_m * (t + y)) / z
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= -0.85)
		tmp = Float64(Float64(x_m / z) * Float64(t + y));
	elseif (z <= 1.0)
		tmp = Float64(x_m * Float64(Float64(y / z) + Float64(-t)));
	else
		tmp = Float64(Float64(x_m * Float64(t + y)) / z);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (z <= -0.85)
		tmp = (x_m / z) * (t + y);
	elseif (z <= 1.0)
		tmp = x_m * ((y / z) + -t);
	else
		tmp = (x_m * (t + y)) / z;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, -0.85], N[(N[(x$95$m / z), $MachinePrecision] * N[(t + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.0], N[(x$95$m * N[(N[(y / z), $MachinePrecision] + (-t)), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * N[(t + y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -0.849999999999999978

   1. Initial program 96.3%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. remove-double-neg N/A

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

      3. neg-mul-1 N/A

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

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

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

      5. neg-mul-1 N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

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

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

      9. *-commutative N/A

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

      10. mul-1-neg N/A

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

      11. lower-/.f64 N/A

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

   5. Applied rewrites 85.2%

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

   7. Applied rewrites 87.9%

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

   if -0.849999999999999978 < z < 1

   1. Initial program 92.4%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. associate-/r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lower-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. distribute-neg-in N/A

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

      14. metadata-eval N/A

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

      15. remove-double-neg N/A

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

      16. lower-+.f64 92.3

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

   4. Applied rewrites 92.3%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 91.8

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

   7. Applied rewrites 91.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 91.8

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. div-inv N/A

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

      10. lower-/.f64 N/A

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

   9. Applied rewrites 91.8%

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

   if 1 < z

   1. Initial program 94.9%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. remove-double-neg N/A

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

      3. neg-mul-1 N/A

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

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

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

      5. neg-mul-1 N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

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

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

      9. *-commutative N/A

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

      10. mul-1-neg N/A

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

      11. lower-/.f64 N/A

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

   5. Applied rewrites 89.3%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.85:\\ \;\;\;\;\frac{x}{z} \cdot \left(t + y\right)\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x \cdot \left(\frac{y}{z} + \left(-t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(t + y\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := x\_m \cdot \frac{t}{z + -1}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -1.35 \cdot 10^{+66}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 330000000000:\\ \;\;\;\;\frac{x\_m \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (* x_m (/ t (+ z -1.0)))))
   (*
    x_s
    (if (<= t -1.35e+66) t_1 (if (<= t 330000000000.0) (/ (* x_m y) z) t_1)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m * (t / (z + -1.0));
	double tmp;
	if (t <= -1.35e+66) {
		tmp = t_1;
	} else if (t <= 330000000000.0) {
		tmp = (x_m * y) / z;
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x_m * (t / (z + (-1.0d0)))
    if (t <= (-1.35d+66)) then
        tmp = t_1
    else if (t <= 330000000000.0d0) then
        tmp = (x_m * y) / z
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m * (t / (z + -1.0));
	double tmp;
	if (t <= -1.35e+66) {
		tmp = t_1;
	} else if (t <= 330000000000.0) {
		tmp = (x_m * y) / z;
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	t_1 = x_m * (t / (z + -1.0))
	tmp = 0
	if t <= -1.35e+66:
		tmp = t_1
	elif t <= 330000000000.0:
		tmp = (x_m * y) / z
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(x_m * Float64(t / Float64(z + -1.0)))
	tmp = 0.0
	if (t <= -1.35e+66)
		tmp = t_1;
	elseif (t <= 330000000000.0)
		tmp = Float64(Float64(x_m * y) / z);
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = x_m * (t / (z + -1.0));
	tmp = 0.0;
	if (t <= -1.35e+66)
		tmp = t_1;
	elseif (t <= 330000000000.0)
		tmp = (x_m * y) / z;
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(x$95$m * N[(t / N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t, -1.35e+66], t$95$1, If[LessEqual[t, 330000000000.0], N[(N[(x$95$m * y), $MachinePrecision] / z), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.35e66 or 3.3e11 < t

   1. Initial program 95.4%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 76.0

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

   5. Applied rewrites 76.0%

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

   if -1.35e66 < t < 3.3e11

   1. Initial program 92.8%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 87.9

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

   5. Applied rewrites 87.9%

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

Alternative 7: 74.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \frac{x\_m}{z} \cdot \left(t + y\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -8.8 \cdot 10^{-162}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 4.7 \cdot 10^{-246}:\\ \;\;\;\;\frac{x\_m \cdot t}{z + -1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (* (/ x_m z) (+ t y))))
   (*
    x_s
    (if (<= y -8.8e-162)
      t_1
      (if (<= y 4.7e-246) (/ (* x_m t) (+ z -1.0)) t_1)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (x_m / z) * (t + y);
	double tmp;
	if (y <= -8.8e-162) {
		tmp = t_1;
	} else if (y <= 4.7e-246) {
		tmp = (x_m * t) / (z + -1.0);
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x_m / z) * (t + y)
    if (y <= (-8.8d-162)) then
        tmp = t_1
    else if (y <= 4.7d-246) then
        tmp = (x_m * t) / (z + (-1.0d0))
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (x_m / z) * (t + y);
	double tmp;
	if (y <= -8.8e-162) {
		tmp = t_1;
	} else if (y <= 4.7e-246) {
		tmp = (x_m * t) / (z + -1.0);
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	t_1 = (x_m / z) * (t + y)
	tmp = 0
	if y <= -8.8e-162:
		tmp = t_1
	elif y <= 4.7e-246:
		tmp = (x_m * t) / (z + -1.0)
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(Float64(x_m / z) * Float64(t + y))
	tmp = 0.0
	if (y <= -8.8e-162)
		tmp = t_1;
	elseif (y <= 4.7e-246)
		tmp = Float64(Float64(x_m * t) / Float64(z + -1.0));
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = (x_m / z) * (t + y);
	tmp = 0.0;
	if (y <= -8.8e-162)
		tmp = t_1;
	elseif (y <= 4.7e-246)
		tmp = (x_m * t) / (z + -1.0);
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x$95$m / z), $MachinePrecision] * N[(t + y), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -8.8e-162], t$95$1, If[LessEqual[y, 4.7e-246], N[(N[(x$95$m * t), $MachinePrecision] / N[(z + -1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8.7999999999999997e-162 or 4.69999999999999955e-246 < y

   1. Initial program 93.4%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. remove-double-neg N/A

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

      3. neg-mul-1 N/A

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

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

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

      5. neg-mul-1 N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

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

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

      9. *-commutative N/A

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

      10. mul-1-neg N/A

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

      11. lower-/.f64 N/A

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

   5. Applied rewrites 79.2%

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

   7. Applied rewrites 80.8%

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

   if -8.7999999999999997e-162 < y < 4.69999999999999955e-246

   1. Initial program 95.9%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 95.9

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

   4. Applied rewrites 95.9%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. distribute-lft-in N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. distribute-frac-neg N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. distribute-neg-in N/A

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

      13. lift-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. frac-2neg N/A

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

      18. div-inv N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lift-+.f64 N/A

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

   6. Applied rewrites 95.9%

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

   7. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 83.9

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

   9. Applied rewrites 83.9%

      \[\leadsto \color{blue}{\frac{x \cdot t}{z + -1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.8 \cdot 10^{-162}:\\ \;\;\;\;\frac{x}{z} \cdot \left(t + y\right)\\ \mathbf{elif}\;y \leq 4.7 \cdot 10^{-246}:\\ \;\;\;\;\frac{x \cdot t}{z + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(t + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := t \cdot \frac{x\_m}{z + -1}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -2.5 \cdot 10^{+85}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 350000000000:\\ \;\;\;\;\frac{x\_m \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (* t (/ x_m (+ z -1.0)))))
   (*
    x_s
    (if (<= t -2.5e+85) t_1 (if (<= t 350000000000.0) (/ (* x_m y) z) t_1)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = t * (x_m / (z + -1.0));
	double tmp;
	if (t <= -2.5e+85) {
		tmp = t_1;
	} else if (t <= 350000000000.0) {
		tmp = (x_m * y) / z;
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t * (x_m / (z + (-1.0d0)))
    if (t <= (-2.5d+85)) then
        tmp = t_1
    else if (t <= 350000000000.0d0) then
        tmp = (x_m * y) / z
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = t * (x_m / (z + -1.0));
	double tmp;
	if (t <= -2.5e+85) {
		tmp = t_1;
	} else if (t <= 350000000000.0) {
		tmp = (x_m * y) / z;
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	t_1 = t * (x_m / (z + -1.0))
	tmp = 0
	if t <= -2.5e+85:
		tmp = t_1
	elif t <= 350000000000.0:
		tmp = (x_m * y) / z
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(t * Float64(x_m / Float64(z + -1.0)))
	tmp = 0.0
	if (t <= -2.5e+85)
		tmp = t_1;
	elseif (t <= 350000000000.0)
		tmp = Float64(Float64(x_m * y) / z);
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = t * (x_m / (z + -1.0));
	tmp = 0.0;
	if (t <= -2.5e+85)
		tmp = t_1;
	elseif (t <= 350000000000.0)
		tmp = (x_m * y) / z;
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(t * N[(x$95$m / N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t, -2.5e+85], t$95$1, If[LessEqual[t, 350000000000.0], N[(N[(x$95$m * y), $MachinePrecision] / z), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.5e85 or 3.5e11 < t

   1. Initial program 95.2%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 31.2

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

   5. Applied rewrites 31.2%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-neg-in N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. distribute-neg-frac2 N/A

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

      9. remove-double-neg N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. lower-+.f64 71.6

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

   8. Applied rewrites 71.6%

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

   if -2.5e85 < t < 3.5e11

   1. Initial program 92.9%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 87.1

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

   5. Applied rewrites 87.1%

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

Alternative 9: 66.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := x\_m \cdot \frac{t}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -1.45 \cdot 10^{+67}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{+109}:\\ \;\;\;\;\frac{x\_m \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (* x_m (/ t z))))
   (* x_s (if (<= t -1.45e+67) t_1 (if (<= t 6.5e+109) (/ (* x_m y) z) t_1)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m * (t / z);
	double tmp;
	if (t <= -1.45e+67) {
		tmp = t_1;
	} else if (t <= 6.5e+109) {
		tmp = (x_m * y) / z;
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x_m * (t / z)
    if (t <= (-1.45d+67)) then
        tmp = t_1
    else if (t <= 6.5d+109) then
        tmp = (x_m * y) / z
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m * (t / z);
	double tmp;
	if (t <= -1.45e+67) {
		tmp = t_1;
	} else if (t <= 6.5e+109) {
		tmp = (x_m * y) / z;
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	t_1 = x_m * (t / z)
	tmp = 0
	if t <= -1.45e+67:
		tmp = t_1
	elif t <= 6.5e+109:
		tmp = (x_m * y) / z
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(x_m * Float64(t / z))
	tmp = 0.0
	if (t <= -1.45e+67)
		tmp = t_1;
	elseif (t <= 6.5e+109)
		tmp = Float64(Float64(x_m * y) / z);
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = x_m * (t / z);
	tmp = 0.0;
	if (t <= -1.45e+67)
		tmp = t_1;
	elseif (t <= 6.5e+109)
		tmp = (x_m * y) / z;
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(x$95$m * N[(t / z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t, -1.45e+67], t$95$1, If[LessEqual[t, 6.5e+109], N[(N[(x$95$m * y), $MachinePrecision] / z), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.45000000000000012e67 or 6.5e109 < t

   1. Initial program 95.4%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 76.9

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

   5. Applied rewrites 76.9%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 50.4%

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

   if -1.45000000000000012e67 < t < 6.5e109

   1. Initial program 93.1%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 82.2

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

   5. Applied rewrites 82.2%

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

Alternative 10: 62.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{-178}:\\ \;\;\;\;\frac{x\_m \cdot y}{z}\\ \mathbf{elif}\;y \leq 4.7 \cdot 10^{-246}:\\ \;\;\;\;x\_m \cdot \left(-t\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x\_m}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= y -8.5e-178)
    (/ (* x_m y) z)
    (if (<= y 4.7e-246) (* x_m (- t)) (* y (/ x_m z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -8.5e-178) {
		tmp = (x_m * y) / z;
	} else if (y <= 4.7e-246) {
		tmp = x_m * -t;
	} else {
		tmp = y * (x_m / z);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-8.5d-178)) then
        tmp = (x_m * y) / z
    else if (y <= 4.7d-246) then
        tmp = x_m * -t
    else
        tmp = y * (x_m / z)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -8.5e-178) {
		tmp = (x_m * y) / z;
	} else if (y <= 4.7e-246) {
		tmp = x_m * -t;
	} else {
		tmp = y * (x_m / z);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if y <= -8.5e-178:
		tmp = (x_m * y) / z
	elif y <= 4.7e-246:
		tmp = x_m * -t
	else:
		tmp = y * (x_m / z)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (y <= -8.5e-178)
		tmp = Float64(Float64(x_m * y) / z);
	elseif (y <= 4.7e-246)
		tmp = Float64(x_m * Float64(-t));
	else
		tmp = Float64(y * Float64(x_m / z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (y <= -8.5e-178)
		tmp = (x_m * y) / z;
	elseif (y <= 4.7e-246)
		tmp = x_m * -t;
	else
		tmp = y * (x_m / z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[y, -8.5e-178], N[(N[(x$95$m * y), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[y, 4.7e-246], N[(x$95$m * (-t)), $MachinePrecision], N[(y * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -8.5000000000000001e-178

   1. Initial program 94.2%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 74.4

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

   5. Applied rewrites 74.4%

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

   if -8.5000000000000001e-178 < y < 4.69999999999999955e-246

   1. Initial program 95.8%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 85.0

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

   5. Applied rewrites 85.0%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 59.5%

      \[\leadsto x \cdot \left(-t\right) \]

   if 4.69999999999999955e-246 < y

   1. Initial program 92.7%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 72.8

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

   5. Applied rewrites 72.8%

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

   7. Applied rewrites 73.9%

      \[\leadsto y \cdot \color{blue}{\frac{x}{z}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 62.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := y \cdot \frac{x\_m}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -8.8 \cdot 10^{-178}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 4.7 \cdot 10^{-246}:\\ \;\;\;\;x\_m \cdot \left(-t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (* y (/ x_m z))))
   (* x_s (if (<= y -8.8e-178) t_1 (if (<= y 4.7e-246) (* x_m (- t)) t_1)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = y * (x_m / z);
	double tmp;
	if (y <= -8.8e-178) {
		tmp = t_1;
	} else if (y <= 4.7e-246) {
		tmp = x_m * -t;
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = y * (x_m / z)
    if (y <= (-8.8d-178)) then
        tmp = t_1
    else if (y <= 4.7d-246) then
        tmp = x_m * -t
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = y * (x_m / z);
	double tmp;
	if (y <= -8.8e-178) {
		tmp = t_1;
	} else if (y <= 4.7e-246) {
		tmp = x_m * -t;
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	t_1 = y * (x_m / z)
	tmp = 0
	if y <= -8.8e-178:
		tmp = t_1
	elif y <= 4.7e-246:
		tmp = x_m * -t
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(y * Float64(x_m / z))
	tmp = 0.0
	if (y <= -8.8e-178)
		tmp = t_1;
	elseif (y <= 4.7e-246)
		tmp = Float64(x_m * Float64(-t));
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = y * (x_m / z);
	tmp = 0.0;
	if (y <= -8.8e-178)
		tmp = t_1;
	elseif (y <= 4.7e-246)
		tmp = x_m * -t;
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -8.8e-178], t$95$1, If[LessEqual[y, 4.7e-246], N[(x$95$m * (-t)), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8.8000000000000005e-178 or 4.69999999999999955e-246 < y

   1. Initial program 93.4%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 73.5

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

   5. Applied rewrites 73.5%

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

   7. Applied rewrites 73.2%

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

   if -8.8000000000000005e-178 < y < 4.69999999999999955e-246

   1. Initial program 95.8%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 85.0

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

   5. Applied rewrites 85.0%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 59.5%

      \[\leadsto x \cdot \left(-t\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 23.7% accurate, 4.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(x\_m \cdot \left(-t\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t) :precision binary64 (* x_s (* x_m (- t))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	return x_s * (x_m * -t);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x_s * (x_m * -t)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	return x_s * (x_m * -t);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	return x_s * (x_m * -t)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	return Float64(x_s * Float64(x_m * Float64(-t)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y, z, t)
	tmp = x_s * (x_m * -t);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * N[(x$95$m * (-t)), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.9%

   \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
2. Add Preprocessing

3. Taylor expanded in y around 0

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. lower-/.f64 N/A

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

   4. sub-neg N/A

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

   5. mul-1-neg N/A

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

   6. +-commutative N/A

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

   7. distribute-neg-in N/A

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

   8. mul-1-neg N/A

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

   9. remove-double-neg N/A

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

   10. metadata-eval N/A

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

   11. lower-+.f64 43.6

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

5. Applied rewrites 43.6%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 23.4%

   \[\leadsto x \cdot \left(-t\right) \]
9. Add Preprocessing

Developer Target 1: 94.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ t_2 := x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\\ \mathbf{if}\;t\_2 < -7.623226303312042 \cdot 10^{-196}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 < 1.4133944927702302 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z} + \left(-\frac{t \cdot x}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (- (/ y z) (* t (/ 1.0 (- 1.0 z))))))
        (t_2 (* x (- (/ y z) (/ t (- 1.0 z))))))
   (if (< t_2 -7.623226303312042e-196)
     t_1
     (if (< t_2 1.4133944927702302e-211)
       (+ (/ (* y x) z) (- (/ (* t x) (- 1.0 z))))
       t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * ((y / z) - (t * (1.0 / (1.0 - z))));
	double t_2 = x * ((y / z) - (t / (1.0 - z)));
	double tmp;
	if (t_2 < -7.623226303312042e-196) {
		tmp = t_1;
	} else if (t_2 < 1.4133944927702302e-211) {
		tmp = ((y * x) / z) + -((t * x) / (1.0 - z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x * ((y / z) - (t * (1.0d0 / (1.0d0 - z))))
    t_2 = x * ((y / z) - (t / (1.0d0 - z)))
    if (t_2 < (-7.623226303312042d-196)) then
        tmp = t_1
    else if (t_2 < 1.4133944927702302d-211) then
        tmp = ((y * x) / z) + -((t * x) / (1.0d0 - z))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * ((y / z) - (t * (1.0 / (1.0 - z))));
	double t_2 = x * ((y / z) - (t / (1.0 - z)));
	double tmp;
	if (t_2 < -7.623226303312042e-196) {
		tmp = t_1;
	} else if (t_2 < 1.4133944927702302e-211) {
		tmp = ((y * x) / z) + -((t * x) / (1.0 - z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * ((y / z) - (t * (1.0 / (1.0 - z))))
	t_2 = x * ((y / z) - (t / (1.0 - z)))
	tmp = 0
	if t_2 < -7.623226303312042e-196:
		tmp = t_1
	elif t_2 < 1.4133944927702302e-211:
		tmp = ((y * x) / z) + -((t * x) / (1.0 - z))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(Float64(y / z) - Float64(t * Float64(1.0 / Float64(1.0 - z)))))
	t_2 = Float64(x * Float64(Float64(y / z) - Float64(t / Float64(1.0 - z))))
	tmp = 0.0
	if (t_2 < -7.623226303312042e-196)
		tmp = t_1;
	elseif (t_2 < 1.4133944927702302e-211)
		tmp = Float64(Float64(Float64(y * x) / z) + Float64(-Float64(Float64(t * x) / Float64(1.0 - z))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * ((y / z) - (t * (1.0 / (1.0 - z))));
	t_2 = x * ((y / z) - (t / (1.0 - z)));
	tmp = 0.0;
	if (t_2 < -7.623226303312042e-196)
		tmp = t_1;
	elseif (t_2 < 1.4133944927702302e-211)
		tmp = ((y * x) / z) + -((t * x) / (1.0 - z));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(N[(y / z), $MachinePrecision] - N[(t * N[(1.0 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(N[(y / z), $MachinePrecision] - N[(t / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[t$95$2, -7.623226303312042e-196], t$95$1, If[Less[t$95$2, 1.4133944927702302e-211], N[(N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision] + (-N[(N[(t * x), $MachinePrecision] / N[(1.0 - z), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], t$95$1]]]]

Reproduce

herbie shell --seed 2024221 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, C"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (* x (- (/ y z) (/ t (- 1 z)))) -3811613151656021/5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (* x (- (/ y z) (* t (/ 1 (- 1 z))))) (if (< (* x (- (/ y z) (/ t (- 1 z)))) 7066972463851151/50000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ (/ (* y x) z) (- (/ (* t x) (- 1 z)))) (* x (- (/ y z) (* t (/ 1 (- 1 z))))))))

  (* x (- (/ y z) (/ t (- 1.0 z)))))