Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2, A

Average Error: 1.9 → 1.9

Time: 28.6s

Precision: binary64

Cost: 20160

Math

\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]

↓

\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t + -1\right) \cdot \log a\right) - b}}{y} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y))

↓

(FPCore (x y z t a b)
 :precision binary64
 (/ (* x (exp (- (+ (* y (log z)) (* (+ t -1.0) (log a))) b))) y))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (x * exp((((y * log(z)) + ((t - 1.0) * log(a))) - b))) / y;
}

↓

double code(double x, double y, double z, double t, double a, double b) {
	return (x * exp((((y * log(z)) + ((t + -1.0) * log(a))) - b))) / y;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (x * exp((((y * log(z)) + ((t - 1.0d0) * log(a))) - b))) / y
end function

↓

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (x * exp((((y * log(z)) + ((t + (-1.0d0)) * log(a))) - b))) / y
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return (x * Math.exp((((y * Math.log(z)) + ((t - 1.0) * Math.log(a))) - b))) / y;
}

↓

public static double code(double x, double y, double z, double t, double a, double b) {
	return (x * Math.exp((((y * Math.log(z)) + ((t + -1.0) * Math.log(a))) - b))) / y;
}

Python

def code(x, y, z, t, a, b):
	return (x * math.exp((((y * math.log(z)) + ((t - 1.0) * math.log(a))) - b))) / y

↓

def code(x, y, z, t, a, b):
	return (x * math.exp((((y * math.log(z)) + ((t + -1.0) * math.log(a))) - b))) / y

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(x * exp(Float64(Float64(Float64(y * log(z)) + Float64(Float64(t - 1.0) * log(a))) - b))) / y)
end

↓

function code(x, y, z, t, a, b)
	return Float64(Float64(x * exp(Float64(Float64(Float64(y * log(z)) + Float64(Float64(t + -1.0) * log(a))) - b))) / y)
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = (x * exp((((y * log(z)) + ((t - 1.0) * log(a))) - b))) / y;
end

↓

function tmp = code(x, y, z, t, a, b)
	tmp = (x * exp((((y * log(z)) + ((t + -1.0) * log(a))) - b))) / y;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(x * N[Exp[N[(N[(N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t - 1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_, b_] := N[(N[(x * N[Exp[N[(N[(N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t + -1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]

Target

Original	1.9
Target	11.2
Herbie	1.9

\[\begin{array}{l} \mathbf{if}\;t < -0.8845848504127471:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{\left(t - 1\right)}}{y}}{\left(b + 1\right) - y \cdot \log z}\\ \mathbf{elif}\;t < 852031.2288374073:\\ \;\;\;\;\frac{\frac{x}{y} \cdot {a}^{\left(t - 1\right)}}{e^{b - \log z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{\left(t - 1\right)}}{y}}{\left(b + 1\right) - y \cdot \log z}\\ \end{array} \]

Derivation

1. Initial program 1.9

   \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]

2. Final simplification 1.9

   \[\leadsto \frac{x \cdot e^{\left(y \cdot \log z + \left(t + -1\right) \cdot \log a\right) - b}}{y} \]

Alternatives

Alternative 1
Error	2.3
Cost	26692

\[\begin{array}{l} \mathbf{if}\;\left(t + -1\right) \cdot \log a \leq -680:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot e^{\left(y \cdot \log z - \log a\right) - b}}{y}\\ \end{array} \]

Alternative 2
Error	12.6
Cost	14100

\[\begin{array}{l} t_1 := a \cdot \left(y \cdot e^{b}\right)\\ t_2 := {a}^{\left(t + -1\right)}\\ t_3 := x \cdot {z}^{y}\\ \mathbf{if}\;b \leq -4 \cdot 10^{+91}:\\ \;\;\;\;\frac{x}{y \cdot \left(a \cdot b\right)}\\ \mathbf{elif}\;b \leq -16:\\ \;\;\;\;\frac{x \cdot t_2}{y}\\ \mathbf{elif}\;b \leq -1.3 \cdot 10^{-72}:\\ \;\;\;\;\frac{t_3}{t_1}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-304}:\\ \;\;\;\;\frac{{a}^{t}}{a} \cdot \frac{x}{y}\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{-119}:\\ \;\;\;\;\frac{x \cdot \frac{t_2}{e^{b}}}{y}\\ \mathbf{elif}\;b \leq 9 \cdot 10^{-13}:\\ \;\;\;\;\frac{t_3}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t_1}\\ \end{array} \]

Alternative 3
Error	8.2
Cost	14034

\[\begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+180} \lor \neg \left(y \leq 10^{+25}\right) \land \left(y \leq 1.85 \cdot 10^{+76} \lor \neg \left(y \leq 1.2 \cdot 10^{+153}\right)\right):\\ \;\;\;\;\frac{x \cdot {z}^{y}}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot e^{\left(t + -1\right) \cdot \log a - b}}{y}\\ \end{array} \]

Alternative 4
Error	11.8
Cost	7308

\[\begin{array}{l} \mathbf{if}\;b \leq -1.04 \cdot 10^{+99}:\\ \;\;\;\;\frac{x}{y \cdot \left(a \cdot b\right)}\\ \mathbf{elif}\;b \leq 3.6 \cdot 10^{-119}:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{elif}\;b \leq 9 \cdot 10^{-13}:\\ \;\;\;\;\frac{x \cdot {z}^{y}}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \end{array} \]

Alternative 5
Error	21.4
Cost	7244

\[\begin{array}{l} \mathbf{if}\;b \leq -2.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{y \cdot \left(a \cdot \left(1 + b\right)\right)}\\ \mathbf{elif}\;b \leq -1.9 \cdot 10^{-233}:\\ \;\;\;\;\frac{1}{a \cdot \frac{y}{x}}\\ \mathbf{elif}\;b \leq -1.1 \cdot 10^{-306}:\\ \;\;\;\;x \cdot \frac{\frac{-b}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \end{array} \]

Alternative 6
Error	21.4
Cost	7244

\[\begin{array}{l} \mathbf{if}\;b \leq -450:\\ \;\;\;\;\frac{x}{y \cdot \left(a \cdot b\right)}\\ \mathbf{elif}\;b \leq -1.25 \cdot 10^{-233}:\\ \;\;\;\;\frac{\frac{\frac{x}{y}}{e^{b}}}{a}\\ \mathbf{elif}\;b \leq -3.8 \cdot 10^{-304}:\\ \;\;\;\;x \cdot \frac{\frac{-b}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \end{array} \]

Alternative 7
Error	12.7
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b \leq -2.75 \cdot 10^{+163}:\\ \;\;\;\;\frac{x}{y \cdot \left(a \cdot b\right)}\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{-11}:\\ \;\;\;\;\frac{{a}^{t}}{a} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \end{array} \]

Alternative 8
Error	11.2
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b \leq -5.5 \cdot 10^{+98}:\\ \;\;\;\;\frac{x}{y \cdot \left(a \cdot b\right)}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \end{array} \]

Alternative 9
Error	34.2
Cost	976

\[\begin{array}{l} \mathbf{if}\;b \leq -15200:\\ \;\;\;\;\frac{x}{y \cdot \left(a \cdot b\right)}\\ \mathbf{elif}\;b \leq -1.85 \cdot 10^{-233}:\\ \;\;\;\;\frac{1}{a \cdot \frac{y}{x}}\\ \mathbf{elif}\;b \leq -8 \cdot 10^{-306}:\\ \;\;\;\;x \cdot \frac{\frac{-b}{a}}{y}\\ \mathbf{elif}\;b \leq 5.3 \cdot 10^{-11}:\\ \;\;\;\;\frac{x}{\frac{y}{\frac{1}{a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot b\right)}\\ \end{array} \]

Alternative 10
Error	34.1
Cost	972

\[\begin{array}{l} \mathbf{if}\;b \leq -1050:\\ \;\;\;\;\frac{x}{y \cdot \left(a \cdot b\right)}\\ \mathbf{elif}\;b \leq -2.8 \cdot 10^{-233}:\\ \;\;\;\;\frac{1}{a \cdot \frac{y}{x}}\\ \mathbf{elif}\;b \leq 1.75 \cdot 10^{-307}:\\ \;\;\;\;x \cdot \frac{\frac{-b}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y + y \cdot b\right)}\\ \end{array} \]

Alternative 11
Error	34.1
Cost	972

\[\begin{array}{l} \mathbf{if}\;b \leq -1.3 \cdot 10^{-6}:\\ \;\;\;\;\frac{x}{y \cdot \left(a \cdot \left(1 + b\right)\right)}\\ \mathbf{elif}\;b \leq -9.3 \cdot 10^{-234}:\\ \;\;\;\;\frac{1}{a \cdot \frac{y}{x}}\\ \mathbf{elif}\;b \leq 7 \cdot 10^{-309}:\\ \;\;\;\;x \cdot \frac{\frac{-b}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y + y \cdot b\right)}\\ \end{array} \]

Alternative 12
Error	33.9
Cost	844

\[\begin{array}{l} t_1 := \frac{x}{y \cdot \left(a \cdot b\right)}\\ \mathbf{if}\;b \leq -26000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -8.2 \cdot 10^{-297}:\\ \;\;\;\;\frac{1}{a \cdot \frac{y}{x}}\\ \mathbf{elif}\;b \leq 5.3 \cdot 10^{-11}:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	34.4
Cost	844

\[\begin{array}{l} \mathbf{if}\;b \leq -4200:\\ \;\;\;\;\frac{x}{y \cdot \left(a \cdot b\right)}\\ \mathbf{elif}\;b \leq -2.6 \cdot 10^{-294}:\\ \;\;\;\;\frac{1}{a \cdot \frac{y}{x}}\\ \mathbf{elif}\;b \leq 5.3 \cdot 10^{-11}:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot b\right)}\\ \end{array} \]

Alternative 14
Error	34.4
Cost	844

\[\begin{array}{l} \mathbf{if}\;b \leq -820:\\ \;\;\;\;\frac{x}{y \cdot \left(a \cdot b\right)}\\ \mathbf{elif}\;b \leq -2.5 \cdot 10^{-297}:\\ \;\;\;\;\frac{1}{a \cdot \frac{y}{x}}\\ \mathbf{elif}\;b \leq 5.3 \cdot 10^{-11}:\\ \;\;\;\;\frac{x}{\frac{y}{\frac{1}{a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot b\right)}\\ \end{array} \]

Alternative 15
Error	38.4
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-99}:\\ \;\;\;\;x \cdot \frac{\frac{1}{y}}{a}\\ \mathbf{elif}\;x \leq 10^{-23}:\\ \;\;\;\;\frac{1}{a \cdot \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \end{array} \]

Alternative 16
Error	39.9
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-99} \lor \neg \left(x \leq 4 \cdot 10^{-73}\right):\\ \;\;\;\;\frac{x}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \end{array} \]

Alternative 17
Error	38.5
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{+61} \lor \neg \left(x \leq 3.8 \cdot 10^{-70}\right):\\ \;\;\;\;\frac{x}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{a}\\ \end{array} \]

Alternative 18
Error	38.6
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{+61}:\\ \;\;\;\;x \cdot \frac{\frac{1}{y}}{a}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-70}:\\ \;\;\;\;\frac{\frac{x}{y}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \end{array} \]

Alternative 19
Error	41.5
Cost	320

\[\frac{x}{y \cdot a} \]

Reproduce

herbie shell --seed 2022364 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (if (< t -0.8845848504127471) (/ (* x (/ (pow a (- t 1.0)) y)) (- (+ b 1.0) (* y (log z)))) (if (< t 852031.2288374073) (/ (* (/ x y) (pow a (- t 1.0))) (exp (- b (* (log z) y)))) (/ (* x (/ (pow a (- t 1.0)) y)) (- (+ b 1.0) (* y (log z))))))

  (/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y))