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

Average Error: 4.7 → 1.6

Time: 11.0s

Precision: binary64

Cost: 1992

Math

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

↓

\[\begin{array}{l} t_1 := \frac{y}{z} - \frac{t}{1 - z}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+279}:\\ \;\;\;\;\frac{y \cdot \left(1 - z\right) - z \cdot t}{\frac{z \cdot \left(1 - z\right)}{x}}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+289}:\\ \;\;\;\;t_1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (/ y z) (/ t (- 1.0 z)))))
   (if (<= t_1 -5e+279)
     (/ (- (* y (- 1.0 z)) (* z t)) (/ (* z (- 1.0 z)) x))
     (if (<= t_1 5e+289) (* t_1 x) (* y (/ x z))))))

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 t_1 = (y / z) - (t / (1.0 - z));
	double tmp;
	if (t_1 <= -5e+279) {
		tmp = ((y * (1.0 - z)) - (z * t)) / ((z * (1.0 - z)) / x);
	} else if (t_1 <= 5e+289) {
		tmp = t_1 * x;
	} else {
		tmp = y * (x / z);
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = (y / z) - (t / (1.0d0 - z))
    if (t_1 <= (-5d+279)) then
        tmp = ((y * (1.0d0 - z)) - (z * t)) / ((z * (1.0d0 - z)) / x)
    else if (t_1 <= 5d+289) then
        tmp = t_1 * x
    else
        tmp = y * (x / z)
    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 t_1 = (y / z) - (t / (1.0 - z));
	double tmp;
	if (t_1 <= -5e+279) {
		tmp = ((y * (1.0 - z)) - (z * t)) / ((z * (1.0 - z)) / x);
	} else if (t_1 <= 5e+289) {
		tmp = t_1 * x;
	} else {
		tmp = y * (x / z);
	}
	return tmp;
}

Python

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

↓

def code(x, y, z, t):
	t_1 = (y / z) - (t / (1.0 - z))
	tmp = 0
	if t_1 <= -5e+279:
		tmp = ((y * (1.0 - z)) - (z * t)) / ((z * (1.0 - z)) / x)
	elif t_1 <= 5e+289:
		tmp = t_1 * x
	else:
		tmp = y * (x / z)
	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)
	t_1 = Float64(Float64(y / z) - Float64(t / Float64(1.0 - z)))
	tmp = 0.0
	if (t_1 <= -5e+279)
		tmp = Float64(Float64(Float64(y * Float64(1.0 - z)) - Float64(z * t)) / Float64(Float64(z * Float64(1.0 - z)) / x));
	elseif (t_1 <= 5e+289)
		tmp = Float64(t_1 * x);
	else
		tmp = Float64(y * Float64(x / z));
	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)
	t_1 = (y / z) - (t / (1.0 - z));
	tmp = 0.0;
	if (t_1 <= -5e+279)
		tmp = ((y * (1.0 - z)) - (z * t)) / ((z * (1.0 - z)) / x);
	elseif (t_1 <= 5e+289)
		tmp = t_1 * x;
	else
		tmp = y * (x / z);
	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_] := Block[{t$95$1 = N[(N[(y / z), $MachinePrecision] - N[(t / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+279], N[(N[(N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / N[(N[(z * N[(1.0 - z), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+289], N[(t$95$1 * x), $MachinePrecision], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	4.7
Target	4.4
Herbie	1.6

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

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < -5.0000000000000002e279

   1. Initial program 42.7

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

   2. Applied egg-rr 0.9

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

   if -5.0000000000000002e279 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < 5.00000000000000031e289

   1. Initial program 1.4

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

   if 5.00000000000000031e289 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z)))

   1. Initial program 49.1

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

   2. Taylor expanded in y around inf 6.8

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

   3. Simplified 55.7

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

      [Start] 6.8	\[ \frac{y \cdot x}{z} \]
      associate-*l/ [<=] 55.7	\[ \color{blue}{\frac{y}{z} \cdot x} \]

   4. Taylor expanded in y around 0 6.8

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

   5. Simplified 6.8

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

      [Start] 6.8	\[ \frac{y \cdot x}{z} \]
      associate-*r/ [<=] 6.8	\[ \color{blue}{y \cdot \frac{x}{z}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 1.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \leq -5 \cdot 10^{+279}:\\ \;\;\;\;\frac{y \cdot \left(1 - z\right) - z \cdot t}{\frac{z \cdot \left(1 - z\right)}{x}}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 5 \cdot 10^{+289}:\\ \;\;\;\;\left(\frac{y}{z} - \frac{t}{1 - z}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.5
Cost	1993

\[\begin{array}{l} t_1 := \frac{y}{z} - \frac{t}{1 - z}\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 5 \cdot 10^{+289}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot x\\ \end{array} \]

Alternative 2
Error	22.4
Cost	1772

\[\begin{array}{l} t_1 := x \cdot \left(\frac{y}{z} - t\right)\\ t_2 := \frac{y \cdot x}{z}\\ t_3 := x \cdot \frac{t}{z}\\ \mathbf{if}\;z \leq -1.22 \cdot 10^{+174}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;z \leq -2.15 \cdot 10^{+60}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq -8 \cdot 10^{-111}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -3.8 \cdot 10^{-281}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{-279}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{-220}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{-110}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 9 \cdot 10^{+21}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+89}:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+238}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;z \leq 8.078 \cdot 10^{+287}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	12.2
Cost	1504

\[\begin{array}{l} t_1 := \frac{x}{z} \cdot \left(y + t\right)\\ t_2 := x \cdot \left(\frac{y}{z} - t\right)\\ t_3 := \frac{y \cdot x}{z}\\ \mathbf{if}\;z \leq -2.15 \cdot 10^{+133}:\\ \;\;\;\;\frac{x \cdot \left(y + t\right)}{z}\\ \mathbf{elif}\;z \leq -3.7:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.1 \cdot 10^{-111}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -8 \cdot 10^{-283}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 8.4 \cdot 10^{-280}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.66 \cdot 10^{-218}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{-110}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	12.1
Cost	1504

\[\begin{array}{l} t_1 := \frac{x}{z} \cdot \left(y + t\right)\\ t_2 := x \cdot \left(\frac{y}{z} - t\right)\\ t_3 := \frac{y \cdot x}{z}\\ \mathbf{if}\;z \leq -4.2 \cdot 10^{+133}:\\ \;\;\;\;\frac{x \cdot \left(y + t\right)}{z}\\ \mathbf{elif}\;z \leq -3.7:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.85 \cdot 10^{-111}:\\ \;\;\;\;\frac{y}{z} \cdot x - t \cdot x\\ \mathbf{elif}\;z \leq -5.2 \cdot 10^{-279}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 7.4 \cdot 10^{-280}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 4.75 \cdot 10^{-219}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{-111}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	26.2
Cost	1376

\[\begin{array}{l} t_1 := t \cdot \frac{x}{z}\\ t_2 := y \cdot \frac{x}{z}\\ t_3 := \frac{y}{z} \cdot x\\ t_4 := x \cdot \frac{t}{z}\\ \mathbf{if}\;z \leq -7.6 \cdot 10^{+168}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq -1.05 \cdot 10^{+58}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;z \leq 2.45 \cdot 10^{-108}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.04 \cdot 10^{-81}:\\ \;\;\;\;t \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 72000000000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+90}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+239}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 8.078 \cdot 10^{+287}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	26.5
Cost	1376

\[\begin{array}{l} t_1 := y \cdot \frac{x}{z}\\ t_2 := \frac{x}{\frac{z}{y}}\\ t_3 := x \cdot \frac{t}{z}\\ \mathbf{if}\;z \leq -5.4 \cdot 10^{+172}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -4.4 \cdot 10^{+57}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{-108}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-81}:\\ \;\;\;\;t \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+24}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+88}:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{+240}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;z \leq 8.078 \cdot 10^{+287}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Error	26.2
Cost	1376

\[\begin{array}{l} t_1 := \frac{x}{\frac{z}{y}}\\ t_2 := x \cdot \frac{t}{z}\\ \mathbf{if}\;z \leq -1.6 \cdot 10^{+168}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -4.8 \cdot 10^{+60}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{-108}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{-81}:\\ \;\;\;\;t \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{+18}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+89}:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{+236}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;z \leq 8.078 \cdot 10^{+287}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	26.4
Cost	1376

\[\begin{array}{l} t_1 := x \cdot \frac{t}{z}\\ \mathbf{if}\;z \leq -2.25 \cdot 10^{+169}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;z \leq -2.4 \cdot 10^{+60}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.22 \cdot 10^{-109}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-81}:\\ \;\;\;\;t \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+14}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+89}:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{+241}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;z \leq 8.078 \cdot 10^{+287}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array} \]

Alternative 9
Error	11.8
Cost	1372

\[\begin{array}{l} t_1 := x \cdot \left(\frac{y}{z} - t\right)\\ t_2 := \frac{y \cdot x}{z}\\ t_3 := \frac{x}{z} \cdot \left(y + t\right)\\ \mathbf{if}\;z \leq -3.7:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq -1.1 \cdot 10^{-111}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -3.4 \cdot 10^{-283}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-280}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.56 \cdot 10^{-221}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{-111}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 10
Error	28.3
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq 2.9 \cdot 10^{-108} \lor \neg \left(z \leq 8.5 \cdot 10^{-81}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(-x\right)\\ \end{array} \]

Alternative 11
Error	23.0
Cost	585

\[\begin{array}{l} \mathbf{if}\;t \leq -2.2 \cdot 10^{+67} \lor \neg \left(t \leq 2.1 \cdot 10^{+116}\right):\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 12
Error	50.4
Cost	256

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

Reproduce

herbie shell --seed 2023031 
(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)))))