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

Percentage Accurate: 94.1% → 97.9%

Time: 9.6s

Alternatives: 14

Speedup: 0.9×

• 20.5% of points produce a very large (infinite) output. You may want to add a precondition.

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: 94.1% 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: 97.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{z} \cdot y\\ t_2 := \frac{y}{z} - \frac{t}{1 - z}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+291}:\\ \;\;\;\;t\_2 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ x z) y)) (t_2 (- (/ y z) (/ t (- 1.0 z)))))
   (if (<= t_2 (- INFINITY)) t_1 (if (<= t_2 4e+291) (* t_2 x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x / z) * y;
	double t_2 = (y / z) - (t / (1.0 - z));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= 4e+291) {
		tmp = t_2 * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x / z) * y;
	double t_2 = (y / z) - (t / (1.0 - z));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_2 <= 4e+291) {
		tmp = t_2 * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x / z) * y
	t_2 = (y / z) - (t / (1.0 - z))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = t_1
	elif t_2 <= 4e+291:
		tmp = t_2 * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x / z) * y)
	t_2 = Float64(Float64(y / z) - Float64(t / Float64(1.0 - z)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= 4e+291)
		tmp = Float64(t_2 * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x / z) * y;
	t_2 = (y / z) - (t / (1.0 - z));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = t_1;
	elseif (t_2 <= 4e+291)
		tmp = t_2 * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y / z), $MachinePrecision] - N[(t / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, 4e+291], N[(t$95$2 * x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z))) < -inf.0 or 3.9999999999999998e291 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z)))

   1. Initial program 64.8%

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

      1. lift--.f64 N/A

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

      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. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. div-inv N/A

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

      7. lower-fma.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 N/A

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

      10. lift--.f64 N/A

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

      11. sub-neg N/A

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

      12. distribute-neg-in N/A

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

      13. metadata-eval N/A

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

      14. remove-double-neg N/A

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

      15. lower-+.f64 64.8

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

   4. Applied rewrites 64.8%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow-1 N/A

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

      5. div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. lift-/.f64 N/A

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

      10. frac-2neg N/A

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

      11. lift-neg.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. div-inv N/A

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

      14. metadata-eval N/A

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

      15. lift-neg.f64 N/A

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

      16. frac-2neg N/A

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

      17. lift-/.f64 N/A

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

      18. *-commutative N/A

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

      19. associate-*l* N/A

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

      20. lower-fma.f64 N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 97.2

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

   6. Applied rewrites 97.2%

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

   7. Taylor expanded in y around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 100.0

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

   9. Applied rewrites 100.0%

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

   if -inf.0 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z))) < 3.9999999999999998e291

   1. Initial program 98.5%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \leq -\infty:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 4 \cdot 10^{+291}:\\ \;\;\;\;\left(\frac{y}{z} - \frac{t}{1 - z}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.2 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{z}, \left(-y\right) \cdot x, \frac{x}{\frac{z - 1}{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, {\left(z - 1\right)}^{-1}, \frac{y}{z}\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x 2.2e-5)
   (fma (/ -1.0 z) (* (- y) x) (/ x (/ (- z 1.0) t)))
   (* (fma t (pow (- z 1.0) -1.0) (/ y z)) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 2.2e-5) {
		tmp = fma((-1.0 / z), (-y * x), (x / ((z - 1.0) / t)));
	} else {
		tmp = fma(t, pow((z - 1.0), -1.0), (y / z)) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= 2.2e-5)
		tmp = fma(Float64(-1.0 / z), Float64(Float64(-y) * x), Float64(x / Float64(Float64(z - 1.0) / t)));
	else
		tmp = Float64(fma(t, (Float64(z - 1.0) ^ -1.0), Float64(y / z)) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, 2.2e-5], N[(N[(-1.0 / z), $MachinePrecision] * N[((-y) * x), $MachinePrecision] + N[(x / N[(N[(z - 1.0), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t * N[Power[N[(z - 1.0), $MachinePrecision], -1.0], $MachinePrecision] + N[(y / z), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.1999999999999999e-5

   1. Initial program 92.3%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   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. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. div-inv N/A

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

      7. lower-fma.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 N/A

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

      10. lift--.f64 N/A

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

      11. sub-neg N/A

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

      12. distribute-neg-in N/A

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

      13. metadata-eval N/A

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

      14. remove-double-neg N/A

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

      15. lower-+.f64 92.3

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

   4. Applied rewrites 92.3%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow-1 N/A

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

      5. div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. lift-/.f64 N/A

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

      10. frac-2neg N/A

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

      11. lift-neg.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. div-inv N/A

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

      14. metadata-eval N/A

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

      15. lift-neg.f64 N/A

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

      16. frac-2neg N/A

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

      17. lift-/.f64 N/A

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

      18. *-commutative N/A

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

      19. associate-*l* N/A

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

      20. lower-fma.f64 N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 95.0

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

   6. Applied rewrites 95.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 95.0

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

   8. Applied rewrites 95.0%

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

   if 2.1999999999999999e-5 < x

   1. Initial program 97.3%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   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. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. div-inv N/A

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

      7. lower-fma.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 N/A

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

      10. lift--.f64 N/A

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

      11. sub-neg N/A

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

      12. distribute-neg-in N/A

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

      13. metadata-eval N/A

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

      14. remove-double-neg N/A

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

      15. lower-+.f64 97.3

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

   4. Applied rewrites 97.3%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{z}, \left(-y\right) \cdot x, \frac{x}{\frac{z - 1}{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, {\left(z - 1\right)}^{-1}, \frac{y}{z}\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.25 \cdot 10^{+28}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{z}, \left(-y\right) \cdot x, \frac{x}{\frac{z - 1}{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{z} - \frac{t}{1 - z}\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x 3.25e+28)
   (fma (/ -1.0 z) (* (- y) x) (/ x (/ (- z 1.0) t)))
   (* (- (/ y z) (/ t (- 1.0 z))) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 3.25e+28) {
		tmp = fma((-1.0 / z), (-y * x), (x / ((z - 1.0) / t)));
	} else {
		tmp = ((y / z) - (t / (1.0 - z))) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= 3.25e+28)
		tmp = fma(Float64(-1.0 / z), Float64(Float64(-y) * x), Float64(x / Float64(Float64(z - 1.0) / t)));
	else
		tmp = Float64(Float64(Float64(y / z) - Float64(t / Float64(1.0 - z))) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, 3.25e+28], N[(N[(-1.0 / z), $MachinePrecision] * N[((-y) * x), $MachinePrecision] + N[(x / N[(N[(z - 1.0), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / z), $MachinePrecision] - N[(t / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.25e28

   1. Initial program 92.6%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   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. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. div-inv N/A

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

      7. lower-fma.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 N/A

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

      10. lift--.f64 N/A

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

      11. sub-neg N/A

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

      12. distribute-neg-in N/A

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

      13. metadata-eval N/A

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

      14. remove-double-neg N/A

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

      15. lower-+.f64 92.6

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

   4. Applied rewrites 92.6%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow-1 N/A

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

      5. div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. lift-/.f64 N/A

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

      10. frac-2neg N/A

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

      11. lift-neg.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. div-inv N/A

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

      14. metadata-eval N/A

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

      15. lift-neg.f64 N/A

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

      16. frac-2neg N/A

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

      17. lift-/.f64 N/A

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

      18. *-commutative N/A

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

      19. associate-*l* N/A

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

      20. lower-fma.f64 N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 95.1

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

   6. Applied rewrites 95.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 95.2

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

   8. Applied rewrites 95.2%

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

   if 3.25e28 < x

   1. Initial program 97.0%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.25 \cdot 10^{+28}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{z}, \left(-y\right) \cdot x, \frac{x}{\frac{z - 1}{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{z} - \frac{t}{1 - z}\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{z}, \left(-y\right) \cdot x, \frac{t}{z - 1} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{z} - \frac{t}{1 - z}\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x 2e-8)
   (fma (/ -1.0 z) (* (- y) x) (* (/ t (- z 1.0)) x))
   (* (- (/ y z) (/ t (- 1.0 z))) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 2e-8) {
		tmp = fma((-1.0 / z), (-y * x), ((t / (z - 1.0)) * x));
	} else {
		tmp = ((y / z) - (t / (1.0 - z))) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= 2e-8)
		tmp = fma(Float64(-1.0 / z), Float64(Float64(-y) * x), Float64(Float64(t / Float64(z - 1.0)) * x));
	else
		tmp = Float64(Float64(Float64(y / z) - Float64(t / Float64(1.0 - z))) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, 2e-8], N[(N[(-1.0 / z), $MachinePrecision] * N[((-y) * x), $MachinePrecision] + N[(N[(t / N[(z - 1.0), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / z), $MachinePrecision] - N[(t / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2e-8

   1. Initial program 92.3%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   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. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. div-inv N/A

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

      7. lower-fma.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 N/A

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

      10. lift--.f64 N/A

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

      11. sub-neg N/A

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

      12. distribute-neg-in N/A

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

      13. metadata-eval N/A

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

      14. remove-double-neg N/A

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

      15. lower-+.f64 92.3

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

   4. Applied rewrites 92.3%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow-1 N/A

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

      5. div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. lift-/.f64 N/A

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

      10. frac-2neg N/A

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

      11. lift-neg.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. div-inv N/A

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

      14. metadata-eval N/A

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

      15. lift-neg.f64 N/A

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

      16. frac-2neg N/A

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

      17. lift-/.f64 N/A

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

      18. *-commutative N/A

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

      19. associate-*l* N/A

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

      20. lower-fma.f64 N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 95.0

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

   6. Applied rewrites 95.0%

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

   if 2e-8 < x

   1. Initial program 97.3%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{z}, \left(-y\right) \cdot x, \frac{t}{z - 1} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{z} - \frac{t}{1 - z}\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t + y}{z} \cdot x\\ \mathbf{if}\;z \leq -1.2 \cdot 10^{-9}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.08 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{z}, \left(-y\right) \cdot x, \left(-t\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ (+ t y) z) x)))
   (if (<= z -1.2e-9)
     t_1
     (if (<= z 1.08e-11) (fma (/ -1.0 z) (* (- y) x) (* (- t) x)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((t + y) / z) * x;
	double tmp;
	if (z <= -1.2e-9) {
		tmp = t_1;
	} else if (z <= 1.08e-11) {
		tmp = fma((-1.0 / z), (-y * x), (-t * x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(t + y) / z) * x)
	tmp = 0.0
	if (z <= -1.2e-9)
		tmp = t_1;
	elseif (z <= 1.08e-11)
		tmp = fma(Float64(-1.0 / z), Float64(Float64(-y) * x), Float64(Float64(-t) * x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(t + y), $MachinePrecision] / z), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[z, -1.2e-9], t$95$1, If[LessEqual[z, 1.08e-11], N[(N[(-1.0 / z), $MachinePrecision] * N[((-y) * x), $MachinePrecision] + N[((-t) * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.2e-9 or 1.07999999999999992e-11 < z

   1. Initial program 97.8%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 96.6

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

   5. Applied rewrites 96.6%

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

   if -1.2e-9 < z < 1.07999999999999992e-11

   1. Initial program 88.5%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   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. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. div-inv N/A

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

      7. lower-fma.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 N/A

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

      10. lift--.f64 N/A

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

      11. sub-neg N/A

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

      12. distribute-neg-in N/A

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

      13. metadata-eval N/A

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

      14. remove-double-neg N/A

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

      15. lower-+.f64 88.5

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

   4. Applied rewrites 88.5%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow-1 N/A

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

      5. div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. lift-/.f64 N/A

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

      10. frac-2neg N/A

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

      11. lift-neg.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. div-inv N/A

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

      14. metadata-eval N/A

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

      15. lift-neg.f64 N/A

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

      16. frac-2neg N/A

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

      17. lift-/.f64 N/A

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

      18. *-commutative N/A

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

      19. associate-*l* N/A

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

      20. lower-fma.f64 N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 96.3

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

   6. Applied rewrites 96.3%

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

   7. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 96.3

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

   9. Applied rewrites 96.3%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{t + y}{z} \cdot x\\ \mathbf{elif}\;z \leq 1.08 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{z}, \left(-y\right) \cdot x, \left(-t\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t + y}{z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 42.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{z} \cdot t\\ t_2 := \mathsf{fma}\left(x, z, x\right) \cdot \left(-t\right)\\ \mathbf{if}\;z \leq -0.75:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{-197}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{-181}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 225000:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ x z) t)) (t_2 (* (fma x z x) (- t))))
   (if (<= z -0.75)
     t_1
     (if (<= z -1.8e-197)
       t_2
       (if (<= z 3.7e-181) t_1 (if (<= z 225000.0) t_2 t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x / z) * t;
	double t_2 = fma(x, z, x) * -t;
	double tmp;
	if (z <= -0.75) {
		tmp = t_1;
	} else if (z <= -1.8e-197) {
		tmp = t_2;
	} else if (z <= 3.7e-181) {
		tmp = t_1;
	} else if (z <= 225000.0) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x / z) * t)
	t_2 = Float64(fma(x, z, x) * Float64(-t))
	tmp = 0.0
	if (z <= -0.75)
		tmp = t_1;
	elseif (z <= -1.8e-197)
		tmp = t_2;
	elseif (z <= 3.7e-181)
		tmp = t_1;
	elseif (z <= 225000.0)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x / z), $MachinePrecision] * t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * z + x), $MachinePrecision] * (-t)), $MachinePrecision]}, If[LessEqual[z, -0.75], t$95$1, If[LessEqual[z, -1.8e-197], t$95$2, If[LessEqual[z, 3.7e-181], t$95$1, If[LessEqual[z, 225000.0], t$95$2, t$95$1]]]]]]

Derivation

1. Split input into 2 regimes

2. if z < -0.75 or -1.7999999999999999e-197 < z < 3.69999999999999984e-181 or 225000 < z

   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 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. remove-double-neg N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 41.6

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

   5. Applied rewrites 41.6%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 48.1%

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

   if -0.75 < z < -1.7999999999999999e-197 or 3.69999999999999984e-181 < z < 225000

   1. Initial program 95.5%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. remove-double-neg N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 41.1

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

   5. Applied rewrites 41.1%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 39.5%

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

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.75:\\ \;\;\;\;\frac{x}{z} \cdot t\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{-197}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right) \cdot \left(-t\right)\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{-181}:\\ \;\;\;\;\frac{x}{z} \cdot t\\ \mathbf{elif}\;z \leq 225000:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right) \cdot \left(-t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{z} \cdot x\\ \mathbf{if}\;z \leq -1.55 \cdot 10^{+264}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;z \leq -6.1 \cdot 10^{+63}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{+25}:\\ \;\;\;\;\left(\frac{y}{z} - t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ t z) x)))
   (if (<= z -1.55e+264)
     (* (/ y z) x)
     (if (<= z -6.1e+63) t_1 (if (<= z 3.4e+25) (* (- (/ y z) t) x) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t / z) * x;
	double tmp;
	if (z <= -1.55e+264) {
		tmp = (y / z) * x;
	} else if (z <= -6.1e+63) {
		tmp = t_1;
	} else if (z <= 3.4e+25) {
		tmp = ((y / z) - t) * x;
	} 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) :: tmp
    t_1 = (t / z) * x
    if (z <= (-1.55d+264)) then
        tmp = (y / z) * x
    else if (z <= (-6.1d+63)) then
        tmp = t_1
    else if (z <= 3.4d+25) then
        tmp = ((y / z) - t) * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (t / z) * x;
	double tmp;
	if (z <= -1.55e+264) {
		tmp = (y / z) * x;
	} else if (z <= -6.1e+63) {
		tmp = t_1;
	} else if (z <= 3.4e+25) {
		tmp = ((y / z) - t) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (t / z) * x
	tmp = 0
	if z <= -1.55e+264:
		tmp = (y / z) * x
	elif z <= -6.1e+63:
		tmp = t_1
	elif z <= 3.4e+25:
		tmp = ((y / z) - t) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t / z) * x)
	tmp = 0.0
	if (z <= -1.55e+264)
		tmp = Float64(Float64(y / z) * x);
	elseif (z <= -6.1e+63)
		tmp = t_1;
	elseif (z <= 3.4e+25)
		tmp = Float64(Float64(Float64(y / z) - t) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (t / z) * x;
	tmp = 0.0;
	if (z <= -1.55e+264)
		tmp = (y / z) * x;
	elseif (z <= -6.1e+63)
		tmp = t_1;
	elseif (z <= 3.4e+25)
		tmp = ((y / z) - t) * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t / z), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[z, -1.55e+264], N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[z, -6.1e+63], t$95$1, If[LessEqual[z, 3.4e+25], N[(N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.54999999999999991e264

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 81.9

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

   5. Applied rewrites 81.9%

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

   if -1.54999999999999991e264 < z < -6.09999999999999968e63 or 3.39999999999999984e25 < z

   1. Initial program 96.8%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 48.3

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

   5. Applied rewrites 48.3%

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

   6. Taylor expanded in z around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 96.8

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

   8. Applied rewrites 96.8%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 66.8%

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

   if -6.09999999999999968e63 < z < 3.39999999999999984e25

   1. Initial program 91.0%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 86.3

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

   5. Applied rewrites 86.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 86.4%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{+264}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;z \leq -6.1 \cdot 10^{+63}:\\ \;\;\;\;\frac{t}{z} \cdot x\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{+25}:\\ \;\;\;\;\left(\frac{y}{z} - t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 95.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t + y}{z} \cdot x\\ \mathbf{if}\;z \leq -0.88:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.08 \cdot 10^{-11}:\\ \;\;\;\;\frac{\left(y - t \cdot z\right) \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ (+ t y) z) x)))
   (if (<= z -0.88) t_1 (if (<= z 1.08e-11) (/ (* (- y (* t z)) x) z) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((t + y) / z) * x;
	double tmp;
	if (z <= -0.88) {
		tmp = t_1;
	} else if (z <= 1.08e-11) {
		tmp = ((y - (t * z)) * x) / 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) :: tmp
    t_1 = ((t + y) / z) * x
    if (z <= (-0.88d0)) then
        tmp = t_1
    else if (z <= 1.08d-11) then
        tmp = ((y - (t * z)) * x) / 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 = ((t + y) / z) * x;
	double tmp;
	if (z <= -0.88) {
		tmp = t_1;
	} else if (z <= 1.08e-11) {
		tmp = ((y - (t * z)) * x) / z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((t + y) / z) * x
	tmp = 0
	if z <= -0.88:
		tmp = t_1
	elif z <= 1.08e-11:
		tmp = ((y - (t * z)) * x) / z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(t + y) / z) * x)
	tmp = 0.0
	if (z <= -0.88)
		tmp = t_1;
	elseif (z <= 1.08e-11)
		tmp = Float64(Float64(Float64(y - Float64(t * z)) * x) / z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((t + y) / z) * x;
	tmp = 0.0;
	if (z <= -0.88)
		tmp = t_1;
	elseif (z <= 1.08e-11)
		tmp = ((y - (t * z)) * x) / z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(t + y), $MachinePrecision] / z), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[z, -0.88], t$95$1, If[LessEqual[z, 1.08e-11], N[(N[(N[(y - N[(t * z), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] / z), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -0.880000000000000004 or 1.07999999999999992e-11 < z

   1. Initial program 97.7%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 97.7

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

   5. Applied rewrites 97.7%

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

   if -0.880000000000000004 < z < 1.07999999999999992e-11

   1. Initial program 89.2%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

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

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

      9. unsub-neg N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. mul-1-neg N/A

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

      13. unsub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-*.f64 94.5

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

   5. Applied rewrites 94.5%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.88:\\ \;\;\;\;\frac{t + y}{z} \cdot x\\ \mathbf{elif}\;z \leq 1.08 \cdot 10^{-11}:\\ \;\;\;\;\frac{\left(y - t \cdot z\right) \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t + y}{z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 92.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t + y}{z} \cdot x\\ \mathbf{if}\;z \leq -1.6 \cdot 10^{+25}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.08 \cdot 10^{-11}:\\ \;\;\;\;\left(\frac{y}{z} - t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ (+ t y) z) x)))
   (if (<= z -1.6e+25) t_1 (if (<= z 1.08e-11) (* (- (/ y z) t) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((t + y) / z) * x;
	double tmp;
	if (z <= -1.6e+25) {
		tmp = t_1;
	} else if (z <= 1.08e-11) {
		tmp = ((y / z) - t) * x;
	} 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) :: tmp
    t_1 = ((t + y) / z) * x
    if (z <= (-1.6d+25)) then
        tmp = t_1
    else if (z <= 1.08d-11) then
        tmp = ((y / z) - t) * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = ((t + y) / z) * x;
	double tmp;
	if (z <= -1.6e+25) {
		tmp = t_1;
	} else if (z <= 1.08e-11) {
		tmp = ((y / z) - t) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((t + y) / z) * x
	tmp = 0
	if z <= -1.6e+25:
		tmp = t_1
	elif z <= 1.08e-11:
		tmp = ((y / z) - t) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(t + y) / z) * x)
	tmp = 0.0
	if (z <= -1.6e+25)
		tmp = t_1;
	elseif (z <= 1.08e-11)
		tmp = Float64(Float64(Float64(y / z) - t) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((t + y) / z) * x;
	tmp = 0.0;
	if (z <= -1.6e+25)
		tmp = t_1;
	elseif (z <= 1.08e-11)
		tmp = ((y / z) - t) * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(t + y), $MachinePrecision] / z), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[z, -1.6e+25], t$95$1, If[LessEqual[z, 1.08e-11], N[(N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.6e25 or 1.07999999999999992e-11 < z

   1. Initial program 97.6%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 97.6

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

   5. Applied rewrites 97.6%

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

   if -1.6e25 < z < 1.07999999999999992e-11

   1. Initial program 89.4%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 88.2

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

   5. Applied rewrites 88.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 88.2%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{+25}:\\ \;\;\;\;\frac{t + y}{z} \cdot x\\ \mathbf{elif}\;z \leq 1.08 \cdot 10^{-11}:\\ \;\;\;\;\left(\frac{y}{z} - t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{t + y}{z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t \cdot x}{z - 1}\\ \mathbf{if}\;t \leq -2.3 \cdot 10^{+123}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 7.6 \cdot 10^{+106}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* t x) (- z 1.0))))
   (if (<= t -2.3e+123) t_1 (if (<= t 7.6e+106) (* (/ x z) y) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t * x) / (z - 1.0);
	double tmp;
	if (t <= -2.3e+123) {
		tmp = t_1;
	} else if (t <= 7.6e+106) {
		tmp = (x / z) * y;
	} 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) :: tmp
    t_1 = (t * x) / (z - 1.0d0)
    if (t <= (-2.3d+123)) then
        tmp = t_1
    else if (t <= 7.6d+106) then
        tmp = (x / z) * y
    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 = (t * x) / (z - 1.0);
	double tmp;
	if (t <= -2.3e+123) {
		tmp = t_1;
	} else if (t <= 7.6e+106) {
		tmp = (x / z) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (t * x) / (z - 1.0)
	tmp = 0
	if t <= -2.3e+123:
		tmp = t_1
	elif t <= 7.6e+106:
		tmp = (x / z) * y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t * x) / Float64(z - 1.0))
	tmp = 0.0
	if (t <= -2.3e+123)
		tmp = t_1;
	elseif (t <= 7.6e+106)
		tmp = Float64(Float64(x / z) * y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (t * x) / (z - 1.0);
	tmp = 0.0;
	if (t <= -2.3e+123)
		tmp = t_1;
	elseif (t <= 7.6e+106)
		tmp = (x / z) * y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t * x), $MachinePrecision] / N[(z - 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.3e+123], t$95$1, If[LessEqual[t, 7.6e+106], N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.2999999999999999e123 or 7.5999999999999996e106 < t

   1. Initial program 96.1%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. remove-double-neg N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 72.8

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

   5. Applied rewrites 72.8%

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

   if -2.2999999999999999e123 < t < 7.5999999999999996e106

   1. Initial program 92.6%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   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. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. div-inv N/A

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

      7. lower-fma.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 N/A

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

      10. lift--.f64 N/A

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

      11. sub-neg N/A

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

      12. distribute-neg-in N/A

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

      13. metadata-eval N/A

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

      14. remove-double-neg N/A

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

      15. lower-+.f64 92.6

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

   4. Applied rewrites 92.6%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow-1 N/A

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

      5. div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. lift-/.f64 N/A

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

      10. frac-2neg N/A

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

      11. lift-neg.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. div-inv N/A

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

      14. metadata-eval N/A

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

      15. lift-neg.f64 N/A

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

      16. frac-2neg N/A

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

      17. lift-/.f64 N/A

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

      18. *-commutative N/A

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

      19. associate-*l* N/A

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

      20. lower-fma.f64 N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 92.2

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

   6. Applied rewrites 92.2%

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

   7. Taylor expanded in y around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 79.3

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

   9. Applied rewrites 79.3%

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

Alternative 11: 68.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{z} \cdot x\\ \mathbf{if}\;t \leq -7.2 \cdot 10^{+131}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{+133}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ t z) x)))
   (if (<= t -7.2e+131) t_1 (if (<= t 2.1e+133) (* (/ x z) y) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t / z) * x;
	double tmp;
	if (t <= -7.2e+131) {
		tmp = t_1;
	} else if (t <= 2.1e+133) {
		tmp = (x / z) * y;
	} 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) :: tmp
    t_1 = (t / z) * x
    if (t <= (-7.2d+131)) then
        tmp = t_1
    else if (t <= 2.1d+133) then
        tmp = (x / z) * y
    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 = (t / z) * x;
	double tmp;
	if (t <= -7.2e+131) {
		tmp = t_1;
	} else if (t <= 2.1e+133) {
		tmp = (x / z) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (t / z) * x
	tmp = 0
	if t <= -7.2e+131:
		tmp = t_1
	elif t <= 2.1e+133:
		tmp = (x / z) * y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t / z) * x)
	tmp = 0.0
	if (t <= -7.2e+131)
		tmp = t_1;
	elseif (t <= 2.1e+133)
		tmp = Float64(Float64(x / z) * y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (t / z) * x;
	tmp = 0.0;
	if (t <= -7.2e+131)
		tmp = t_1;
	elseif (t <= 2.1e+133)
		tmp = (x / z) * y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t / z), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t, -7.2e+131], t$95$1, If[LessEqual[t, 2.1e+133], N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -7.20000000000000063e131 or 2.1e133 < t

   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 inf

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

      1. lower-/.f64 18.5

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

   5. Applied rewrites 18.5%

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

   6. Taylor expanded in z around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 73.6

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

   8. Applied rewrites 73.6%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 66.2%

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

   if -7.20000000000000063e131 < t < 2.1e133

   1. Initial program 92.8%

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

      1. lift--.f64 N/A

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

      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. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. div-inv N/A

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

      7. lower-fma.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 N/A

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

      10. lift--.f64 N/A

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

      11. sub-neg N/A

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

      12. distribute-neg-in N/A

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

      13. metadata-eval N/A

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

      14. remove-double-neg N/A

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

      15. lower-+.f64 92.8

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

   4. Applied rewrites 92.8%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow-1 N/A

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

      5. div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. lift-/.f64 N/A

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

      10. frac-2neg N/A

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

      11. lift-neg.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. div-inv N/A

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

      14. metadata-eval N/A

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

      15. lift-neg.f64 N/A

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

      16. frac-2neg N/A

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

      17. lift-/.f64 N/A

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

      18. *-commutative N/A

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

      19. associate-*l* N/A

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

      20. lower-fma.f64 N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 92.5

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

   6. Applied rewrites 92.5%

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

   7. Taylor expanded in y around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 78.4

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

   9. Applied rewrites 78.4%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.2 \cdot 10^{+131}:\\ \;\;\;\;\frac{t}{z} \cdot x\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{+133}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 65.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{z} \cdot t\\ \mathbf{if}\;t \leq -7.2 \cdot 10^{+131}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{+133}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ x z) t)))
   (if (<= t -7.2e+131) t_1 (if (<= t 2.1e+133) (* (/ x z) y) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x / z) * t;
	double tmp;
	if (t <= -7.2e+131) {
		tmp = t_1;
	} else if (t <= 2.1e+133) {
		tmp = (x / z) * y;
	} 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) :: tmp
    t_1 = (x / z) * t
    if (t <= (-7.2d+131)) then
        tmp = t_1
    else if (t <= 2.1d+133) then
        tmp = (x / z) * y
    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 / z) * t;
	double tmp;
	if (t <= -7.2e+131) {
		tmp = t_1;
	} else if (t <= 2.1e+133) {
		tmp = (x / z) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x / z) * t
	tmp = 0
	if t <= -7.2e+131:
		tmp = t_1
	elif t <= 2.1e+133:
		tmp = (x / z) * y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x / z) * t)
	tmp = 0.0
	if (t <= -7.2e+131)
		tmp = t_1;
	elseif (t <= 2.1e+133)
		tmp = Float64(Float64(x / z) * y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x / z) * t;
	tmp = 0.0;
	if (t <= -7.2e+131)
		tmp = t_1;
	elseif (t <= 2.1e+133)
		tmp = (x / z) * y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x / z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t, -7.2e+131], t$95$1, If[LessEqual[t, 2.1e+133], N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -7.20000000000000063e131 or 2.1e133 < t

   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

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. remove-double-neg N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 73.1

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

   5. Applied rewrites 73.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 56.3%

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

   if -7.20000000000000063e131 < t < 2.1e133

   1. Initial program 92.8%

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

      1. lift--.f64 N/A

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

      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. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. div-inv N/A

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

      7. lower-fma.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 N/A

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

      10. lift--.f64 N/A

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

      11. sub-neg N/A

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

      12. distribute-neg-in N/A

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

      13. metadata-eval N/A

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

      14. remove-double-neg N/A

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

      15. lower-+.f64 92.8

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

   4. Applied rewrites 92.8%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow-1 N/A

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

      5. div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. lift-/.f64 N/A

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

      10. frac-2neg N/A

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

      11. lift-neg.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. div-inv N/A

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

      14. metadata-eval N/A

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

      15. lift-neg.f64 N/A

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

      16. frac-2neg N/A

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

      17. lift-/.f64 N/A

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

      18. *-commutative N/A

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

      19. associate-*l* N/A

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

      20. lower-fma.f64 N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 92.5

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

   6. Applied rewrites 92.5%

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

   7. Taylor expanded in y around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 78.4

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

   9. Applied rewrites 78.4%

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

Alternative 13: 63.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{z} \cdot t\\ \mathbf{if}\;t \leq -5.2 \cdot 10^{+131}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 13000000:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ x z) t)))
   (if (<= t -5.2e+131) t_1 (if (<= t 13000000.0) (/ (* y x) z) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x / z) * t;
	double tmp;
	if (t <= -5.2e+131) {
		tmp = t_1;
	} else if (t <= 13000000.0) {
		tmp = (y * x) / 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) :: tmp
    t_1 = (x / z) * t
    if (t <= (-5.2d+131)) then
        tmp = t_1
    else if (t <= 13000000.0d0) then
        tmp = (y * x) / 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 / z) * t;
	double tmp;
	if (t <= -5.2e+131) {
		tmp = t_1;
	} else if (t <= 13000000.0) {
		tmp = (y * x) / z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x / z) * t
	tmp = 0
	if t <= -5.2e+131:
		tmp = t_1
	elif t <= 13000000.0:
		tmp = (y * x) / z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x / z) * t)
	tmp = 0.0
	if (t <= -5.2e+131)
		tmp = t_1;
	elseif (t <= 13000000.0)
		tmp = Float64(Float64(y * x) / z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x / z) * t;
	tmp = 0.0;
	if (t <= -5.2e+131)
		tmp = t_1;
	elseif (t <= 13000000.0)
		tmp = (y * x) / z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x / z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t, -5.2e+131], t$95$1, If[LessEqual[t, 13000000.0], N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -5.2e131 or 1.3e7 < t

   1. Initial program 96.0%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. remove-double-neg N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 65.2

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

   5. Applied rewrites 65.2%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 54.1%

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

   if -5.2e131 < t < 1.3e7

   1. Initial program 92.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-sub N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. cancel-sign-sub-inv N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-/.f64 82.3

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

   4. Applied rewrites 82.3%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 74.7%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 82.3%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.2 \cdot 10^{+131}:\\ \;\;\;\;\frac{x}{z} \cdot t\\ \mathbf{elif}\;t \leq 13000000:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 22.7% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

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 = -t * x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.6%

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

3. Taylor expanded in z around 0

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

   1. lower-/.f64 N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. lower--.f64 N/A

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

   5. lower-*.f64 63.5

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

5. Applied rewrites 63.5%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 21.3%

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

9. Final simplification 21.3%

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

Developer Target 1: 94.4% 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 2024276 
(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)))))