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

Average Error: 7.18% → 1.55%

Time: 12.5s

Precision: binary64

Cost: 3409

Math

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

↓

\[\begin{array}{l} t_1 := \frac{y}{z} - \frac{t}{1 - z}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+200}:\\ \;\;\;\;\frac{y \cdot \left(1 - z\right) - z \cdot t}{\frac{z \cdot \left(1 - z\right)}{x}}\\ \mathbf{elif}\;t_1 \leq -5 \cdot 10^{-194} \lor \neg \left(t_1 \leq 10^{-153}\right) \land t_1 \leq 2 \cdot 10^{+195}:\\ \;\;\;\;t_1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z} - \frac{t \cdot x}{1 - z}\\ \end{array} \]

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+200)
     (/ (- (* y (- 1.0 z)) (* z t)) (/ (* z (- 1.0 z)) x))
     (if (or (<= t_1 -5e-194) (and (not (<= t_1 1e-153)) (<= t_1 2e+195)))
       (* t_1 x)
       (- (/ (* y x) z) (/ (* t x) (- 1.0 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+200) {
		tmp = ((y * (1.0 - z)) - (z * t)) / ((z * (1.0 - z)) / x);
	} else if ((t_1 <= -5e-194) || (!(t_1 <= 1e-153) && (t_1 <= 2e+195))) {
		tmp = t_1 * x;
	} else {
		tmp = ((y * x) / z) - ((t * x) / (1.0 - 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+200)) then
        tmp = ((y * (1.0d0 - z)) - (z * t)) / ((z * (1.0d0 - z)) / x)
    else if ((t_1 <= (-5d-194)) .or. (.not. (t_1 <= 1d-153)) .and. (t_1 <= 2d+195)) then
        tmp = t_1 * x
    else
        tmp = ((y * x) / z) - ((t * x) / (1.0d0 - 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+200) {
		tmp = ((y * (1.0 - z)) - (z * t)) / ((z * (1.0 - z)) / x);
	} else if ((t_1 <= -5e-194) || (!(t_1 <= 1e-153) && (t_1 <= 2e+195))) {
		tmp = t_1 * x;
	} else {
		tmp = ((y * x) / z) - ((t * x) / (1.0 - 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+200:
		tmp = ((y * (1.0 - z)) - (z * t)) / ((z * (1.0 - z)) / x)
	elif (t_1 <= -5e-194) or (not (t_1 <= 1e-153) and (t_1 <= 2e+195)):
		tmp = t_1 * x
	else:
		tmp = ((y * x) / z) - ((t * x) / (1.0 - 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+200)
		tmp = Float64(Float64(Float64(y * Float64(1.0 - z)) - Float64(z * t)) / Float64(Float64(z * Float64(1.0 - z)) / x));
	elseif ((t_1 <= -5e-194) || (!(t_1 <= 1e-153) && (t_1 <= 2e+195)))
		tmp = Float64(t_1 * x);
	else
		tmp = Float64(Float64(Float64(y * x) / z) - Float64(Float64(t * x) / Float64(1.0 - 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+200)
		tmp = ((y * (1.0 - z)) - (z * t)) / ((z * (1.0 - z)) / x);
	elseif ((t_1 <= -5e-194) || (~((t_1 <= 1e-153)) && (t_1 <= 2e+195)))
		tmp = t_1 * x;
	else
		tmp = ((y * x) / z) - ((t * x) / (1.0 - 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+200], 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[Or[LessEqual[t$95$1, -5e-194], And[N[Not[LessEqual[t$95$1, 1e-153]], $MachinePrecision], LessEqual[t$95$1, 2e+195]]], N[(t$95$1 * x), $MachinePrecision], N[(N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision] - N[(N[(t * x), $MachinePrecision] / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	7.18%
Target	6.56%
Herbie	1.55%

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

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < -5.00000000000000019e200

   1. Initial program 30.94

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

   2. Applied egg-rr 8.83

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

   if -5.00000000000000019e200 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < -5.0000000000000002e-194 or 1.00000000000000004e-153 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < 1.99999999999999995e195

   1. Initial program 0.35

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

   if -5.0000000000000002e-194 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < 1.00000000000000004e-153 or 1.99999999999999995e195 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z)))

   1. Initial program 16.99

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

   2. Taylor expanded in y around 0 2.23

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

4. Final simplification 1.55

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \leq -5 \cdot 10^{+200}:\\ \;\;\;\;\frac{y \cdot \left(1 - z\right) - z \cdot t}{\frac{z \cdot \left(1 - z\right)}{x}}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq -5 \cdot 10^{-194} \lor \neg \left(\frac{y}{z} - \frac{t}{1 - z} \leq 10^{-153}\right) \land \frac{y}{z} - \frac{t}{1 - z} \leq 2 \cdot 10^{+195}:\\ \;\;\;\;\left(\frac{y}{z} - \frac{t}{1 - z}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z} - \frac{t \cdot x}{1 - z}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.98%
Cost	3410

\[\begin{array}{l} t_1 := \frac{y}{z} - \frac{t}{1 - z}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+200} \lor \neg \left(t_1 \leq -5 \cdot 10^{-194} \lor \neg \left(t_1 \leq 10^{-153}\right) \land t_1 \leq 2 \cdot 10^{+195}\right):\\ \;\;\;\;\frac{y \cdot x}{z} - \frac{t \cdot x}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot x\\ \end{array} \]

Alternative 2
Error	1.16%
Cost	3280

\[\begin{array}{l} t_1 := \frac{y}{z} - \frac{t}{1 - z}\\ t_2 := t_1 \cdot x\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;t_1 \leq -5 \cdot 10^{-194}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 4 \cdot 10^{-213}:\\ \;\;\;\;\frac{y \cdot x + t \cdot x}{z}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+287}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 3
Error	32.96%
Cost	1508

\[\begin{array}{l} t_1 := x \cdot \left(\frac{y}{z} - t\right)\\ t_2 := x \cdot \frac{t}{z}\\ t_3 := \frac{y}{z} \cdot x\\ \mathbf{if}\;z \leq -2.7 \cdot 10^{+258}:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq -2.2 \cdot 10^{+190}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;z \leq -2.5 \cdot 10^{+128}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -4.7 \cdot 10^{+83}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq -27000000000000:\\ \;\;\;\;\frac{t \cdot x}{z}\\ \mathbf{elif}\;z \leq -1.35 \cdot 10^{-180}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-276}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;z \leq 75000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.75 \cdot 10^{+196}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 4
Error	41.85%
Cost	1244

\[\begin{array}{l} t_1 := x \cdot \frac{t}{z}\\ t_2 := \frac{y}{z} \cdot x\\ t_3 := \frac{y \cdot x}{z}\\ \mathbf{if}\;z \leq -1.35 \cdot 10^{+258}:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq -1.3 \cdot 10^{+190}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq -3.5 \cdot 10^{+128}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.05 \cdot 10^{+84}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1.15 \cdot 10^{-8}:\\ \;\;\;\;\frac{t \cdot x}{z}\\ \mathbf{elif}\;z \leq 5000000000:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 5.3 \cdot 10^{+196}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	16.52%
Cost	1240

\[\begin{array}{l} t_1 := \frac{x}{z} \cdot \left(y + t\right)\\ t_2 := x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{if}\;z \leq -1.05 \cdot 10^{+240}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2.1 \cdot 10^{+197}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;z \leq -0.96:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -3.5 \cdot 10^{-180}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-272}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;z \leq 48:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	16.63%
Cost	1240

\[\begin{array}{l} t_1 := \frac{x}{z} \cdot \left(y + t\right)\\ t_2 := x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{if}\;z \leq -1.7 \cdot 10^{+245}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2.1 \cdot 10^{+197}:\\ \;\;\;\;\frac{1}{\frac{z}{y \cdot x}}\\ \mathbf{elif}\;z \leq -1.1:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -8.5 \cdot 10^{-181}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-277}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;z \leq 48:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	15.57%
Cost	1108

\[\begin{array}{l} t_1 := x \cdot \left(\frac{y}{z} - t\right)\\ t_2 := \frac{x}{z} \cdot \left(y + t\right)\\ \mathbf{if}\;z \leq -6.8 \cdot 10^{+251}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1:\\ \;\;\;\;\frac{x \cdot \left(y + t\right)}{z}\\ \mathbf{elif}\;z \leq -1.5 \cdot 10^{-179}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-275}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;z \leq 48:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 8
Error	44.25%
Cost	982

\[\begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{+258} \lor \neg \left(z \leq -3.4 \cdot 10^{+190}\right) \land \left(z \leq -2.1 \cdot 10^{+128} \lor \neg \left(z \leq 2000000000\right) \land z \leq 3.05 \cdot 10^{+196}\right):\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 9
Error	9.65%
Cost	976

\[\begin{array}{l} t_1 := x \cdot \left(\frac{y}{z} - t\right)\\ t_2 := x \cdot \frac{y + t}{z}\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -8.5 \cdot 10^{-181}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-272}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 10
Error	36.79%
Cost	585

\[\begin{array}{l} \mathbf{if}\;t \leq -3 \cdot 10^{+95} \lor \neg \left(t \leq 8.8 \cdot 10^{+133}\right):\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 11
Error	35.29%
Cost	585

\[\begin{array}{l} \mathbf{if}\;t \leq -1.8 \cdot 10^{+95} \lor \neg \left(t \leq 9 \cdot 10^{+130}\right):\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array} \]

Alternative 12
Error	35.24%
Cost	584

\[\begin{array}{l} \mathbf{if}\;t \leq -5.8 \cdot 10^{+95}:\\ \;\;\;\;\frac{x}{\frac{z}{t}}\\ \mathbf{elif}\;t \leq 7.8 \cdot 10^{+119}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \end{array} \]

Alternative 13
Error	35.07%
Cost	584

\[\begin{array}{l} \mathbf{if}\;t \leq -1.45 \cdot 10^{+95}:\\ \;\;\;\;\frac{x}{\frac{z}{t}}\\ \mathbf{elif}\;t \leq 3.45 \cdot 10^{+121}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \end{array} \]

Alternative 14
Error	67.47%
Cost	320

\[t \cdot \frac{x}{z} \]

Reproduce

herbie shell --seed 2023088 
(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)))))