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

Average Error: 4.7 → 1.4

Time: 19.4s

Precision: binary64

Cost: 2184

Math

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

↓

\[\begin{array}{l} t_1 := \frac{y}{z} - \frac{t}{1 - z}\\ t_2 := \left(-\frac{t \cdot x}{1 - z}\right) + \frac{y \cdot x}{z}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+240}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+193}:\\ \;\;\;\;x \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

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 (+ (- (/ (* t x) (- 1.0 z))) (/ (* y x) z))))
   (if (<= t_1 -1e+240) t_2 (if (<= t_1 2e+193) (* x t_1) 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 = -((t * x) / (1.0 - z)) + ((y * x) / z);
	double tmp;
	if (t_1 <= -1e+240) {
		tmp = t_2;
	} else if (t_1 <= 2e+193) {
		tmp = x * t_1;
	} 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 = -((t * x) / (1.0d0 - z)) + ((y * x) / z)
    if (t_1 <= (-1d+240)) then
        tmp = t_2
    else if (t_1 <= 2d+193) then
        tmp = x * t_1
    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 = -((t * x) / (1.0 - z)) + ((y * x) / z);
	double tmp;
	if (t_1 <= -1e+240) {
		tmp = t_2;
	} else if (t_1 <= 2e+193) {
		tmp = x * t_1;
	} 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 = -((t * x) / (1.0 - z)) + ((y * x) / z)
	tmp = 0
	if t_1 <= -1e+240:
		tmp = t_2
	elif t_1 <= 2e+193:
		tmp = x * t_1
	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(Float64(t * x) / Float64(1.0 - z))) + Float64(Float64(y * x) / z))
	tmp = 0.0
	if (t_1 <= -1e+240)
		tmp = t_2;
	elseif (t_1 <= 2e+193)
		tmp = Float64(x * t_1);
	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 = -((t * x) / (1.0 - z)) + ((y * x) / z);
	tmp = 0.0;
	if (t_1 <= -1e+240)
		tmp = t_2;
	elseif (t_1 <= 2e+193)
		tmp = x * t_1;
	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[(t * x), $MachinePrecision] / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]) + N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+240], t$95$2, If[LessEqual[t$95$1, 2e+193], N[(x * t$95$1), $MachinePrecision], t$95$2]]]]

Target

Original	4.7
Target	4.3
Herbie	1.4

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < -1.00000000000000001e240 or 2.00000000000000013e193 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z)))

   1. Initial program 21.4

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

   2. Taylor expanded in y around 0 1.1

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

   3. Simplified 1.1

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

      [Start] 1.1	\[ \frac{y \cdot x}{z} + -1 \cdot \frac{t \cdot x}{1 - z} \]
      rational.json-simplify-1 [=>] 1.1	\[ \color{blue}{-1 \cdot \frac{t \cdot x}{1 - z} + \frac{y \cdot x}{z}} \]
      rational.json-simplify-2 [=>] 1.1	\[ \color{blue}{\frac{t \cdot x}{1 - z} \cdot -1} + \frac{y \cdot x}{z} \]
      rational.json-simplify-9 [=>] 1.1	\[ \color{blue}{\left(-\frac{t \cdot x}{1 - z}\right)} + \frac{y \cdot x}{z} \]

   if -1.00000000000000001e240 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < 2.00000000000000013e193

   1. Initial program 1.5

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

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \leq -1 \cdot 10^{+240}:\\ \;\;\;\;\left(-\frac{t \cdot x}{1 - z}\right) + \frac{y \cdot x}{z}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 2 \cdot 10^{+193}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{t \cdot x}{1 - z}\right) + \frac{y \cdot x}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.7
Cost	1992

\[\begin{array}{l} t_1 := \frac{y}{z} - \frac{t}{1 - z}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+202}:\\ \;\;\;\;x \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z} - t \cdot x\\ \end{array} \]

Alternative 2
Error	20.3
Cost	1112

\[\begin{array}{l} t_1 := x \cdot \frac{t}{z}\\ t_2 := x \cdot \frac{y}{z}\\ t_3 := x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{if}\;z \leq -3.3 \cdot 10^{+15}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-305}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 10^{-153}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+254}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{+297}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	6.2
Cost	1104

\[\begin{array}{l} t_1 := x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;x \cdot \frac{y - \left(-t\right)}{z}\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-305}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-151}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{t}{z} + \frac{y}{z}\right) \cdot x\\ \end{array} \]

Alternative 4
Error	6.2
Cost	1040

\[\begin{array}{l} t_1 := x \cdot \left(\frac{y}{z} - t\right)\\ t_2 := x \cdot \frac{y - \left(-t\right)}{z}\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-305}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-153}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	10.5
Cost	976

\[\begin{array}{l} t_1 := x \cdot \left(\frac{y}{z} - t\right)\\ t_2 := \frac{\left(y + t\right) \cdot x}{z}\\ \mathbf{if}\;z \leq -0.98:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-305}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-152}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Error	3.6
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -0.95:\\ \;\;\;\;x \cdot \frac{y - \left(-t\right)}{z}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{x \cdot y}{z} - t \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{t}{z} + \frac{y}{z}\right) \cdot x\\ \end{array} \]

Alternative 7
Error	22.8
Cost	716

\[\begin{array}{l} t_1 := x \cdot \frac{t}{z}\\ \mathbf{if}\;t \leq -3.6 \cdot 10^{+117}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -3.15 \cdot 10^{-189}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{+154}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	33.2
Cost	584

\[\begin{array}{l} t_1 := x \cdot \frac{t}{z}\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 150000:\\ \;\;\;\;x \cdot \left(-t\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	22.6
Cost	584

\[\begin{array}{l} t_1 := x \cdot \frac{t}{z}\\ \mathbf{if}\;t \leq -2.6 \cdot 10^{+120}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{+154}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	50.3
Cost	256

\[x \cdot \left(-t\right) \]

Reproduce

herbie shell --seed 2023077 
(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)))))