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

Average Error: 4.7 → 1.6

Time: 5.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \frac{y}{z} - \frac{t}{1 - z}\\ t_2 := \frac{y \cdot x}{z} - \frac{t \cdot x}{1 - z}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+195}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+193}:\\ \;\;\;\;t_1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

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))))
        (t_2 (- (/ (* y x) z) (/ (* t x) (- 1.0 z)))))
   (if (<= t_1 -1e+195) t_2 (if (<= t_1 2e+193) (* t_1 x) t_2))))

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 t_2 = ((y * x) / z) - ((t * x) / (1.0 - z));
	double tmp;
	if (t_1 <= -1e+195) {
		tmp = t_2;
	} else if (t_1 <= 2e+193) {
		tmp = t_1 * x;
	} else {
		tmp = t_2;
	}
	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) :: tmp
    t_1 = (y / z) - (t / (1.0d0 - z))
    t_2 = ((y * x) / z) - ((t * x) / (1.0d0 - z))
    if (t_1 <= (-1d+195)) then
        tmp = t_2
    else if (t_1 <= 2d+193) then
        tmp = t_1 * x
    else
        tmp = t_2
    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 t_2 = ((y * x) / z) - ((t * x) / (1.0 - z));
	double tmp;
	if (t_1 <= -1e+195) {
		tmp = t_2;
	} else if (t_1 <= 2e+193) {
		tmp = t_1 * x;
	} else {
		tmp = t_2;
	}
	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))
	t_2 = ((y * x) / z) - ((t * x) / (1.0 - z))
	tmp = 0
	if t_1 <= -1e+195:
		tmp = t_2
	elif t_1 <= 2e+193:
		tmp = t_1 * x
	else:
		tmp = t_2
	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)))
	t_2 = Float64(Float64(Float64(y * x) / z) - Float64(Float64(t * x) / Float64(1.0 - z)))
	tmp = 0.0
	if (t_1 <= -1e+195)
		tmp = t_2;
	elseif (t_1 <= 2e+193)
		tmp = Float64(t_1 * x);
	else
		tmp = t_2;
	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));
	t_2 = ((y * x) / z) - ((t * x) / (1.0 - z));
	tmp = 0.0;
	if (t_1 <= -1e+195)
		tmp = t_2;
	elseif (t_1 <= 2e+193)
		tmp = t_1 * x;
	else
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision] - N[(N[(t * x), $MachinePrecision] / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+195], t$95$2, If[LessEqual[t$95$1, 2e+193], N[(t$95$1 * x), $MachinePrecision], t$95$2]]]]

Target

Original	4.7
Target	4.3
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 2 regimes

2. if (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < -9.99999999999999977e194 or 2.00000000000000013e193 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z)))

   1. Initial program 17.4

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

   2. Applied egg-rr 17.4

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

   3. Applied egg-rr 17.8

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(\sqrt[3]{\frac{y}{z}}, {\left(\sqrt[3]{\frac{y}{z}}\right)}^{2}, -\frac{t}{1 - z}\right)} \]

   4. Taylor expanded in t around 0 1.5

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

   if -9.99999999999999977e194 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < 2.00000000000000013e193

   1. Initial program 1.7

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

4. Final simplification 1.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \leq -1 \cdot 10^{+195}:\\ \;\;\;\;\frac{y \cdot x}{z} - \frac{t \cdot x}{1 - z}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 2 \cdot 10^{+193}:\\ \;\;\;\;\left(\frac{y}{z} - \frac{t}{1 - z}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z} - \frac{t \cdot x}{1 - z}\\ \end{array} \]

Reproduce

herbie shell --seed 2022210 
(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)))))