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

Average Error: 4.9 → 0.6

Time: 12.4s

Precision: binary64

Cost: 3280

Math

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

↓

\[\begin{array}{l} t_1 := \frac{y \cdot x}{z}\\ t_2 := \frac{y}{z} - \frac{t}{1 - z}\\ t_3 := x \cdot t_2\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq -5 \cdot 10^{-198}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{-300}:\\ \;\;\;\;\frac{\left(y + t\right) \cdot x}{z}\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+272}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (- (/ y z) (/ t (- 1.0 z)))) (t_3 (* x t_2)))
   (if (<= t_2 (- INFINITY))
     t_1
     (if (<= t_2 -5e-198)
       t_3
       (if (<= t_2 2e-300)
         (/ (* (+ y t) x) z)
         (if (<= t_2 2e+272) t_3 t_1))))))

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 = (y / z) - (t / (1.0 - z));
	double t_3 = x * t_2;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= -5e-198) {
		tmp = t_3;
	} else if (t_2 <= 2e-300) {
		tmp = ((y + t) * x) / z;
	} else if (t_2 <= 2e+272) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

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 = (y / z) - (t / (1.0 - z));
	double t_3 = x * t_2;
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_2 <= -5e-198) {
		tmp = t_3;
	} else if (t_2 <= 2e-300) {
		tmp = ((y + t) * x) / z;
	} else if (t_2 <= 2e+272) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	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 = (y / z) - (t / (1.0 - z))
	t_3 = x * t_2
	tmp = 0
	if t_2 <= -math.inf:
		tmp = t_1
	elif t_2 <= -5e-198:
		tmp = t_3
	elif t_2 <= 2e-300:
		tmp = ((y + t) * x) / z
	elif t_2 <= 2e+272:
		tmp = t_3
	else:
		tmp = t_1
	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(Float64(y / z) - Float64(t / Float64(1.0 - z)))
	t_3 = Float64(x * t_2)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= -5e-198)
		tmp = t_3;
	elseif (t_2 <= 2e-300)
		tmp = Float64(Float64(Float64(y + t) * x) / z);
	elseif (t_2 <= 2e+272)
		tmp = t_3;
	else
		tmp = t_1;
	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 = (y / z) - (t / (1.0 - z));
	t_3 = x * t_2;
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = t_1;
	elseif (t_2 <= -5e-198)
		tmp = t_3;
	elseif (t_2 <= 2e-300)
		tmp = ((y + t) * x) / z;
	elseif (t_2 <= 2e+272)
		tmp = t_3;
	else
		tmp = t_1;
	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[(N[(y / z), $MachinePrecision] - N[(t / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(x * t$95$2), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, -5e-198], t$95$3, If[LessEqual[t$95$2, 2e-300], N[(N[(N[(y + t), $MachinePrecision] * x), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$2, 2e+272], t$95$3, t$95$1]]]]]]]

Target

Original	4.9
Target	4.5
Herbie	0.6

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

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < -inf.0 or 2.0000000000000001e272 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z)))

   1. Initial program 48.9

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

   2. Taylor expanded in y around inf 4.9

      \[\leadsto \color{blue}{\frac{y \cdot x}{z}} \]

   if -inf.0 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < -4.9999999999999999e-198 or 2.00000000000000005e-300 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < 2.0000000000000001e272

   1. Initial program 0.2

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

   if -4.9999999999999999e-198 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < 2.00000000000000005e-300

   1. Initial program 12.2

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

   2. Taylor expanded in z around inf 12.9

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

   3. Simplified 12.9

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

      [Start] 12.9	\[ x \cdot \frac{y - -1 \cdot t}{z} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 12.9	\[ x \cdot \frac{y - \color{blue}{t \cdot -1}}{z} \]
      rational_best_oopsla_all_46_json_45_simplify-92 [=>] 12.9	\[ x \cdot \frac{y - \color{blue}{\left(-t\right)}}{z} \]

   4. Taylor expanded in x around 0 1.1

      \[\leadsto \color{blue}{\frac{\left(y + t\right) \cdot x}{z}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \leq -\infty:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq -5 \cdot 10^{-198}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 2 \cdot 10^{-300}:\\ \;\;\;\;\frac{\left(y + t\right) \cdot x}{z}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 2 \cdot 10^{+272}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	26.1
Cost	848

\[\begin{array}{l} t_1 := x \cdot \frac{y}{z}\\ t_2 := x \cdot \frac{t}{z}\\ \mathbf{if}\;y \leq -9.2 \cdot 10^{-122}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-135}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-108}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-46}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	20.6
Cost	716

\[\begin{array}{l} t_1 := x \cdot \frac{t}{z}\\ \mathbf{if}\;z \leq -1.8 \cdot 10^{+115}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{+184}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	5.7
Cost	712

\[\begin{array}{l} t_1 := x \cdot \frac{y + t}{z}\\ \mathbf{if}\;z \leq -29000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	33.6
Cost	584

\[\begin{array}{l} t_1 := x \cdot \frac{t}{z}\\ \mathbf{if}\;z \leq -29000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \left(-t\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	22.3
Cost	584

\[\begin{array}{l} t_1 := x \cdot \frac{t}{z}\\ \mathbf{if}\;t \leq -1.22 \cdot 10^{+69}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 6.1 \cdot 10^{+117}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	50.5
Cost	256

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

Reproduce

herbie shell --seed 2023090 
(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)))))