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

Average Error: 4.8 → 2.1

Time: 4.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \frac{y \cdot x}{z}\\ t_2 := \frac{t}{1 - z}\\ t_3 := \frac{y}{z} - t_2\\ \mathbf{if}\;t_3 \leq -5 \cdot 10^{+225}:\\ \;\;\;\;t_1 - t \cdot \frac{1}{\frac{1 - z}{x}}\\ \mathbf{elif}\;t_3 \leq -5 \cdot 10^{-215}:\\ \;\;\;\;\frac{y}{z} \cdot x - t_2 \cdot x\\ \mathbf{elif}\;t_3 \leq 2 \cdot 10^{-156}:\\ \;\;\;\;t_1 - \frac{t \cdot x}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot x\\ \end{array} \]

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 x) z)) (t_2 (/ t (- 1.0 z))) (t_3 (- (/ y z) t_2)))
   (if (<= t_3 -5e+225)
     (- t_1 (* t (/ 1.0 (/ (- 1.0 z) x))))
     (if (<= t_3 -5e-215)
       (- (* (/ y z) x) (* t_2 x))
       (if (<= t_3 2e-156) (- t_1 (/ (* t x) (- 1.0 z))) (* t_3 x))))))

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 * x) / z;
	double t_2 = t / (1.0 - z);
	double t_3 = (y / z) - t_2;
	double tmp;
	if (t_3 <= -5e+225) {
		tmp = t_1 - (t * (1.0 / ((1.0 - z) / x)));
	} else if (t_3 <= -5e-215) {
		tmp = ((y / z) * x) - (t_2 * x);
	} else if (t_3 <= 2e-156) {
		tmp = t_1 - ((t * x) / (1.0 - z));
	} else {
		tmp = t_3 * x;
	}
	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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = (y * x) / z
    t_2 = t / (1.0d0 - z)
    t_3 = (y / z) - t_2
    if (t_3 <= (-5d+225)) then
        tmp = t_1 - (t * (1.0d0 / ((1.0d0 - z) / x)))
    else if (t_3 <= (-5d-215)) then
        tmp = ((y / z) * x) - (t_2 * x)
    else if (t_3 <= 2d-156) then
        tmp = t_1 - ((t * x) / (1.0d0 - z))
    else
        tmp = t_3 * x
    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 * x) / z;
	double t_2 = t / (1.0 - z);
	double t_3 = (y / z) - t_2;
	double tmp;
	if (t_3 <= -5e+225) {
		tmp = t_1 - (t * (1.0 / ((1.0 - z) / x)));
	} else if (t_3 <= -5e-215) {
		tmp = ((y / z) * x) - (t_2 * x);
	} else if (t_3 <= 2e-156) {
		tmp = t_1 - ((t * x) / (1.0 - z));
	} else {
		tmp = t_3 * x;
	}
	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 * x) / z
	t_2 = t / (1.0 - z)
	t_3 = (y / z) - t_2
	tmp = 0
	if t_3 <= -5e+225:
		tmp = t_1 - (t * (1.0 / ((1.0 - z) / x)))
	elif t_3 <= -5e-215:
		tmp = ((y / z) * x) - (t_2 * x)
	elif t_3 <= 2e-156:
		tmp = t_1 - ((t * x) / (1.0 - z))
	else:
		tmp = t_3 * x
	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 * x) / z)
	t_2 = Float64(t / Float64(1.0 - z))
	t_3 = Float64(Float64(y / z) - t_2)
	tmp = 0.0
	if (t_3 <= -5e+225)
		tmp = Float64(t_1 - Float64(t * Float64(1.0 / Float64(Float64(1.0 - z) / x))));
	elseif (t_3 <= -5e-215)
		tmp = Float64(Float64(Float64(y / z) * x) - Float64(t_2 * x));
	elseif (t_3 <= 2e-156)
		tmp = Float64(t_1 - Float64(Float64(t * x) / Float64(1.0 - z)));
	else
		tmp = Float64(t_3 * x);
	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 * x) / z;
	t_2 = t / (1.0 - z);
	t_3 = (y / z) - t_2;
	tmp = 0.0;
	if (t_3 <= -5e+225)
		tmp = t_1 - (t * (1.0 / ((1.0 - z) / x)));
	elseif (t_3 <= -5e-215)
		tmp = ((y / z) * x) - (t_2 * x);
	elseif (t_3 <= 2e-156)
		tmp = t_1 - ((t * x) / (1.0 - z));
	else
		tmp = t_3 * x;
	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 * x), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$2 = N[(t / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(y / z), $MachinePrecision] - t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, -5e+225], N[(t$95$1 - N[(t * N[(1.0 / N[(N[(1.0 - z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -5e-215], N[(N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision] - N[(t$95$2 * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e-156], N[(t$95$1 - N[(N[(t * x), $MachinePrecision] / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 * x), $MachinePrecision]]]]]]]

Target

Original	4.8
Target	4.4
Herbie	2.1

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

Derivation

1. Split input into 4 regimes

2. if (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < -4.99999999999999981e225

   1. Initial program 24.6

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

   2. Taylor expanded in y around 0 0.7

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

   3. Applied egg-rr 0.7

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

   if -4.99999999999999981e225 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < -4.99999999999999956e-215

   1. Initial program 0.2

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

   2. Applied egg-rr 1.3

      \[\leadsto \color{blue}{{\left(\sqrt[3]{x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)}\right)}^{3}} \]

   3. Applied egg-rr 0.2

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

   if -4.99999999999999956e-215 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < 2.00000000000000008e-156

   1. Initial program 7.7

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

   2. Taylor expanded in y around 0 1.3

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

   if 2.00000000000000008e-156 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z)))

   1. Initial program 4.3

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 2.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \leq -5 \cdot 10^{+225}:\\ \;\;\;\;\frac{y \cdot x}{z} - t \cdot \frac{1}{\frac{1 - z}{x}}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq -5 \cdot 10^{-215}:\\ \;\;\;\;\frac{y}{z} \cdot x - \frac{t}{1 - z} \cdot x\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 2 \cdot 10^{-156}:\\ \;\;\;\;\frac{y \cdot x}{z} - \frac{t \cdot x}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{z} - \frac{t}{1 - z}\right) \cdot x\\ \end{array} \]

Reproduce

herbie shell --seed 2022192 
(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)))))