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

Percentage Accurate: 94.5% → 94.2%

Time: 9.1s

Precision: binary64

Cost: 836

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq 3.9 \cdot 10^{+149}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(y + t\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= y 3.9e+149) (* x (- (/ y z) (/ t (- 1.0 z)))) (* (/ x z) (+ y t))))

C

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

↓

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 3.9e+149) {
		tmp = x * ((y / z) - (t / (1.0 - z)));
	} else {
		tmp = (x / z) * (y + t);
	}
	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
    code = x * ((y / z) - (t / (1.0d0 - z)))
end function

↓

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) :: tmp
    if (y <= 3.9d+149) then
        tmp = x * ((y / z) - (t / (1.0d0 - z)))
    else
        tmp = (x / z) * (y + t)
    end if
    code = tmp
end function

Java

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

↓

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 3.9e+149) {
		tmp = x * ((y / z) - (t / (1.0 - z)));
	} else {
		tmp = (x / z) * (y + t);
	}
	return tmp;
}

Python

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

↓

def code(x, y, z, t):
	tmp = 0
	if y <= 3.9e+149:
		tmp = x * ((y / z) - (t / (1.0 - z)))
	else:
		tmp = (x / z) * (y + t)
	return tmp

Julia

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

↓

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 3.9e+149)
		tmp = Float64(x * Float64(Float64(y / z) - Float64(t / Float64(1.0 - z))));
	else
		tmp = Float64(Float64(x / z) * Float64(y + t));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 3.9e+149)
		tmp = x * ((y / z) - (t / (1.0 - z)));
	else
		tmp = (x / z) * (y + t);
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_, z_, t_] := If[LessEqual[y, 3.9e+149], N[(x * N[(N[(y / z), $MachinePrecision] - N[(t / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * N[(y + t), $MachinePrecision]), $MachinePrecision]]

Target

Original	94.5%
Target	95.0%
Herbie	94.2%

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

Derivation

1. Split input into 2 regimes

2. if y < 3.8999999999999999e149

   1. Initial program 96.8%

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

   if 3.8999999999999999e149 < y

   1. Initial program 84.4%

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

   2. Taylor expanded in z around inf 76.8%

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

   3. Simplified 97.3%

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

      [Start] 76.8%	\[ \frac{\left(y - -1 \cdot t\right) \cdot x}{z} \]
      *-commutative [<=] 76.8%	\[ \frac{\color{blue}{x \cdot \left(y - -1 \cdot t\right)}}{z} \]
      associate-/l* [=>] 85.2%	\[ \color{blue}{\frac{x}{\frac{z}{y - -1 \cdot t}}} \]
      associate-/r/ [=>] 97.3%	\[ \color{blue}{\frac{x}{z} \cdot \left(y - -1 \cdot t\right)} \]
      cancel-sign-sub-inv [=>] 97.3%	\[ \frac{x}{z} \cdot \color{blue}{\left(y + \left(--1\right) \cdot t\right)} \]
      metadata-eval [=>] 97.3%	\[ \frac{x}{z} \cdot \left(y + \color{blue}{1} \cdot t\right) \]
      *-lft-identity [=>] 97.3%	\[ \frac{x}{z} \cdot \left(y + \color{blue}{t}\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.9 \cdot 10^{+149}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(y + t\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	94.2%
Cost	836

\[\begin{array}{l} \mathbf{if}\;y \leq 3.9 \cdot 10^{+149}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(y + t\right)\\ \end{array} \]

Alternative 2
Accuracy	74.1%
Cost	716

\[\begin{array}{l} t_1 := \frac{x}{\frac{z}{t}}\\ \mathbf{if}\;z \leq -2.3 \cdot 10^{+17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{+28}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{+58}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

Alternative 3
Accuracy	93.7%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;x \cdot \frac{y + t}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \end{array} \]

Alternative 4
Accuracy	93.6%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -1.15 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;\frac{x}{\frac{z}{y + t}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \end{array} \]

Alternative 5
Accuracy	73.1%
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -5600000:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;y \leq 430000:\\ \;\;\;\;x \cdot \frac{t}{z + -1}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 6
Accuracy	91.6%
Cost	712

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

Alternative 7
Accuracy	42.5%
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 2.55 \cdot 10^{-8}\right):\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-t\right)\\ \end{array} \]

Alternative 8
Accuracy	44.6%
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 2.55 \cdot 10^{-8}\right):\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-t\right)\\ \end{array} \]

Alternative 9
Accuracy	66.4%
Cost	584

\[\begin{array}{l} \mathbf{if}\;t \leq -8.5 \cdot 10^{+96}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{+185}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-t\right)\\ \end{array} \]

Alternative 10
Accuracy	67.2%
Cost	584

\[\begin{array}{l} \mathbf{if}\;t \leq -1.7 \cdot 10^{+97}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{+183}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-t\right)\\ \end{array} \]

Alternative 11
Accuracy	67.4%
Cost	584

\[\begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{+97}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{+183}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-t\right)\\ \end{array} \]

Alternative 12
Accuracy	22.9%
Cost	256

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

Reproduce

herbie shell --seed 2023165 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, C"
  :precision binary64

  :herbie-target
  (if (< (* x (- (/ y z) (/ t (- 1.0 z)))) -7.623226303312042e-196) (* x (- (/ y z) (* t (/ 1.0 (- 1.0 z))))) (if (< (* x (- (/ y z) (/ t (- 1.0 z)))) 1.4133944927702302e-211) (+ (/ (* y x) z) (- (/ (* t x) (- 1.0 z)))) (* x (- (/ y z) (* t (/ 1.0 (- 1.0 z)))))))

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