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

Average Error: 1.9 → 1.9

Time: 37.1s

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	11.4
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	2.1
Cost	26692

\[\begin{array}{l} \mathbf{if}\;\left(t + -1\right) \cdot \log a \leq -640:\\ \;\;\;\;\frac{x}{{a}^{\left(1 - t\right)} \cdot \left(y + y \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot e^{\left(y \cdot \log z - \log a\right) - b}}{y}\\ \end{array} \]

Alternative 2
Error	9.9
Cost	14164

\[\begin{array}{l} t_1 := \frac{\frac{\frac{{z}^{y}}{a}}{e^{b}}}{\frac{y}{x}}\\ \mathbf{if}\;b \leq -3.5 \cdot 10^{-39}:\\ \;\;\;\;\frac{x}{{a}^{\left(1 - t\right)} \cdot \left(y + y \cdot b\right)}\\ \mathbf{elif}\;b \leq -9 \cdot 10^{-147}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -1.05 \cdot 10^{-220}:\\ \;\;\;\;\frac{\left(x \cdot {a}^{\left(t + -1\right)}\right) \cdot \left(1 - b\right)}{y}\\ \mathbf{elif}\;b \leq -3.5 \cdot 10^{-257}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 0.7:\\ \;\;\;\;\frac{{z}^{y}}{y} \cdot \left({a}^{t} \cdot \frac{x}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a \cdot e^{b}}}{y}\\ \end{array} \]

Alternative 3
Error	10.1
Cost	14164

\[\begin{array}{l} t_1 := \frac{\frac{\frac{{z}^{y}}{a}}{e^{b}}}{\frac{y}{x}}\\ \mathbf{if}\;b \leq -4.6 \cdot 10^{-39}:\\ \;\;\;\;\frac{x}{{a}^{\left(1 - t\right)} \cdot \left(y + y \cdot b\right)}\\ \mathbf{elif}\;b \leq -2.9 \cdot 10^{-157}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -1.55 \cdot 10^{-220}:\\ \;\;\;\;\frac{\left(x \cdot {a}^{\left(t + -1\right)}\right) \cdot \left(1 - b\right)}{y}\\ \mathbf{elif}\;b \leq -1.8 \cdot 10^{-256}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-29}:\\ \;\;\;\;\frac{{z}^{y}}{y} \cdot \left({a}^{t} \cdot \frac{x}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot e^{\left(t + -1\right) \cdot \log a - b}}{y}\\ \end{array} \]

Alternative 4
Error	10.9
Cost	13968

\[\begin{array}{l} t_1 := \frac{\frac{\frac{{z}^{y}}{a}}{e^{b}}}{\frac{y}{x}}\\ t_2 := \frac{x}{{a}^{\left(1 - t\right)} \cdot \left(y + y \cdot b\right)}\\ \mathbf{if}\;b \leq -4.3 \cdot 10^{-39}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq -1.75 \cdot 10^{-147}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -1.05 \cdot 10^{-220}:\\ \;\;\;\;\frac{\left(x \cdot {a}^{\left(t + -1\right)}\right) \cdot \left(1 - b\right)}{y}\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{-231}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 0.82:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a \cdot e^{b}}}{y}\\ \end{array} \]

Alternative 5
Error	9.9
Cost	13836

\[\begin{array}{l} t_1 := a \cdot e^{b}\\ \mathbf{if}\;b \leq -3.4 \cdot 10^{-46}:\\ \;\;\;\;\frac{x}{{a}^{\left(1 - t\right)} \cdot \left(y + y \cdot b\right)}\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{-74}:\\ \;\;\;\;\frac{\left(x \cdot {a}^{\left(t + -1\right)}\right) \cdot \left(1 - b\right)}{y}\\ \mathbf{elif}\;b \leq 1:\\ \;\;\;\;\frac{x \cdot {z}^{y}}{y \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t_1}}{y}\\ \end{array} \]

Alternative 6
Error	10.1
Cost	7564

\[\begin{array}{l} t_1 := \frac{x}{{a}^{\left(1 - t\right)} \cdot \left(y + y \cdot b\right)}\\ \mathbf{if}\;b \leq -1.45 \cdot 10^{-220}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 3.6 \cdot 10^{-231}:\\ \;\;\;\;x \cdot \frac{\frac{{z}^{y}}{a}}{y}\\ \mathbf{elif}\;b \leq 0.8:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a \cdot e^{b}}}{y}\\ \end{array} \]

Alternative 7
Error	21.3
Cost	7508

\[\begin{array}{l} t_1 := \frac{x}{a} \cdot \left(-\frac{b}{y}\right)\\ \mathbf{if}\;b \leq -1.4 \cdot 10^{-203}:\\ \;\;\;\;\frac{x}{a \cdot \left(y + y \cdot b\right)}\\ \mathbf{elif}\;b \leq -6.5 \cdot 10^{-272}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 2.3 \cdot 10^{-287}:\\ \;\;\;\;\frac{\frac{a - a \cdot b}{a \cdot a}}{\frac{y}{x}}\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{-266}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 0.92:\\ \;\;\;\;\frac{x - x \cdot b}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a \cdot e^{b}}}{y}\\ \end{array} \]

Alternative 8
Error	11.0
Cost	7440

\[\begin{array}{l} t_1 := x \cdot \frac{\frac{{z}^{y}}{a}}{y}\\ t_2 := x \cdot \frac{{a}^{\left(t + -1\right)}}{y}\\ \mathbf{if}\;b \leq -5.2 \cdot 10^{-221}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 6.6 \cdot 10^{-231}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{-74}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 330:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a \cdot e^{b}}}{y}\\ \end{array} \]

Alternative 9
Error	10.0
Cost	7432

\[\begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{-46}:\\ \;\;\;\;\frac{x}{{a}^{\left(1 - t\right)} \cdot \left(y + y \cdot b\right)}\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-73}:\\ \;\;\;\;\frac{\left(x \cdot {a}^{\left(t + -1\right)}\right) \cdot \left(1 - b\right)}{y}\\ \mathbf{elif}\;b \leq 55:\\ \;\;\;\;x \cdot \frac{\frac{{z}^{y}}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a \cdot e^{b}}}{y}\\ \end{array} \]

Alternative 10
Error	24.2
Cost	7376

\[\begin{array}{l} t_1 := \frac{x}{a} \cdot \left(-\frac{b}{y}\right)\\ \mathbf{if}\;b \leq -4.5 \cdot 10^{-205}:\\ \;\;\;\;\frac{x}{a \cdot \left(y + y \cdot b\right)}\\ \mathbf{elif}\;b \leq -3.5 \cdot 10^{-272}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-287}:\\ \;\;\;\;\frac{\frac{a - a \cdot b}{a \cdot a}}{\frac{y}{x}}\\ \mathbf{elif}\;b \leq 1.85 \cdot 10^{-266}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{e^{b}}}{y \cdot a}\\ \end{array} \]

Alternative 11
Error	10.7
Cost	7308

\[\begin{array}{l} t_1 := \frac{x}{\frac{y}{\frac{{a}^{t}}{a}}}\\ \mathbf{if}\;b \leq -7.6 \cdot 10^{-221}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{-231}:\\ \;\;\;\;x \cdot \frac{\frac{{z}^{y}}{a}}{y}\\ \mathbf{elif}\;b \leq 0.41:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a \cdot e^{b}}}{y}\\ \end{array} \]

Alternative 12
Error	13.3
Cost	7044

\[\begin{array}{l} \mathbf{if}\;b \leq 185:\\ \;\;\;\;x \cdot \frac{\frac{{z}^{y}}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a \cdot e^{b}}}{y}\\ \end{array} \]

Alternative 13
Error	41.9
Cost	1368

\[\begin{array}{l} t_1 := \frac{x}{a} \cdot \left(-\frac{b}{y}\right)\\ \mathbf{if}\;b \leq -5.5 \cdot 10^{-206}:\\ \;\;\;\;\frac{\frac{1}{a}}{\frac{y}{x}}\\ \mathbf{elif}\;b \leq -4 \cdot 10^{-270}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -3.5 \cdot 10^{-303}:\\ \;\;\;\;\frac{\frac{x}{y}}{a}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-269}:\\ \;\;\;\;\frac{-\frac{b}{a}}{\frac{y}{x}}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{-266}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 0.84:\\ \;\;\;\;\left(1 - b\right) \cdot \frac{x}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{\frac{x}{a}}}\\ \end{array} \]

Alternative 14
Error	42.0
Cost	1240

\[\begin{array}{l} t_1 := \frac{x}{a} \cdot \left(-\frac{b}{y}\right)\\ \mathbf{if}\;b \leq -4.9 \cdot 10^{-206}:\\ \;\;\;\;\frac{\frac{1}{a}}{\frac{y}{x}}\\ \mathbf{elif}\;b \leq -9.4 \cdot 10^{-271}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -3 \cdot 10^{-303}:\\ \;\;\;\;\frac{\frac{x}{y}}{a}\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{-269}:\\ \;\;\;\;\frac{-\frac{b}{a}}{\frac{y}{x}}\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{-266}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{+37}:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{\frac{x}{a}}}\\ \end{array} \]

Alternative 15
Error	34.8
Cost	1228

\[\begin{array}{l} t_1 := \frac{x}{a} \cdot \left(-\frac{b}{y}\right)\\ t_2 := \frac{x}{a \cdot \left(y + y \cdot b\right)}\\ \mathbf{if}\;b \leq -6.2 \cdot 10^{-203}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq -2.05 \cdot 10^{-271}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{-287}:\\ \;\;\;\;\frac{\frac{a - a \cdot b}{a \cdot a}}{\frac{y}{x}}\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-266}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 16
Error	40.3
Cost	908

\[\begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{1}{a}}{\frac{y}{x}}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-32}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{+236}:\\ \;\;\;\;\frac{x}{a} \cdot \left(-\frac{b}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \end{array} \]

Alternative 17
Error	34.6
Cost	841

\[\begin{array}{l} \mathbf{if}\;b \leq -3.2 \cdot 10^{-205} \lor \neg \left(b \leq 1.3 \cdot 10^{-266}\right):\\ \;\;\;\;\frac{x}{a \cdot \left(y + y \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a} \cdot \left(-\frac{b}{y}\right)\\ \end{array} \]

Alternative 18
Error	40.5
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{-119} \lor \neg \left(x \leq 2 \cdot 10^{-80}\right):\\ \;\;\;\;\frac{x}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \end{array} \]

Alternative 19
Error	39.0
Cost	580

\[\begin{array}{l} \mathbf{if}\;a \leq 2.8 \cdot 10^{-74}:\\ \;\;\;\;\frac{1}{a} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \end{array} \]

Alternative 20
Error	38.8
Cost	580

\[\begin{array}{l} \mathbf{if}\;a \leq 2.8 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{1}{a}}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \end{array} \]

Alternative 21
Error	39.0
Cost	452

\[\begin{array}{l} \mathbf{if}\;a \leq 2.8 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{x}{y}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \end{array} \]

Alternative 22
Error	42.0
Cost	320

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

Reproduce

herbie shell --seed 2023060 
(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))