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

Average Error: 4.7 → 0.2

Time: 12.4s

Precision: binary64

Cost: 3280

Math

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

↓

\[\begin{array}{l} t_1 := \frac{y}{\frac{z}{x}}\\ t_2 := \frac{y}{z} - \frac{t}{1 - z}\\ t_3 := t_2 \cdot x\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq -1 \cdot 10^{-258}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{-314}:\\ \;\;\;\;\frac{\left(y + t\right) + \frac{t}{z}}{\frac{z}{x}}\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+307}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (/ z x))) (t_2 (- (/ y z) (/ t (- 1.0 z)))) (t_3 (* t_2 x)))
   (if (<= t_2 (- INFINITY))
     t_1
     (if (<= t_2 -1e-258)
       t_3
       (if (<= t_2 5e-314)
         (/ (+ (+ y t) (/ t z)) (/ z x))
         (if (<= t_2 2e+307) 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 / (z / x);
	double t_2 = (y / z) - (t / (1.0 - z));
	double t_3 = t_2 * x;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= -1e-258) {
		tmp = t_3;
	} else if (t_2 <= 5e-314) {
		tmp = ((y + t) + (t / z)) / (z / x);
	} else if (t_2 <= 2e+307) {
		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 / (z / x);
	double t_2 = (y / z) - (t / (1.0 - z));
	double t_3 = t_2 * x;
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_2 <= -1e-258) {
		tmp = t_3;
	} else if (t_2 <= 5e-314) {
		tmp = ((y + t) + (t / z)) / (z / x);
	} else if (t_2 <= 2e+307) {
		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 / (z / x)
	t_2 = (y / z) - (t / (1.0 - z))
	t_3 = t_2 * x
	tmp = 0
	if t_2 <= -math.inf:
		tmp = t_1
	elif t_2 <= -1e-258:
		tmp = t_3
	elif t_2 <= 5e-314:
		tmp = ((y + t) + (t / z)) / (z / x)
	elif t_2 <= 2e+307:
		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(y / Float64(z / x))
	t_2 = Float64(Float64(y / z) - Float64(t / Float64(1.0 - z)))
	t_3 = Float64(t_2 * x)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= -1e-258)
		tmp = t_3;
	elseif (t_2 <= 5e-314)
		tmp = Float64(Float64(Float64(y + t) + Float64(t / z)) / Float64(z / x));
	elseif (t_2 <= 2e+307)
		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 / (z / x);
	t_2 = (y / z) - (t / (1.0 - z));
	t_3 = t_2 * x;
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = t_1;
	elseif (t_2 <= -1e-258)
		tmp = t_3;
	elseif (t_2 <= 5e-314)
		tmp = ((y + t) + (t / z)) / (z / x);
	elseif (t_2 <= 2e+307)
		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[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y / z), $MachinePrecision] - N[(t / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * x), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, -1e-258], t$95$3, If[LessEqual[t$95$2, 5e-314], N[(N[(N[(y + t), $MachinePrecision] + N[(t / z), $MachinePrecision]), $MachinePrecision] / N[(z / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+307], t$95$3, t$95$1]]]]]]]

Target

Original	4.7
Target	4.4
Herbie	0.2

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

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < -inf.0 or 1.99999999999999997e307 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z)))

   1. Initial program 63.0

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

   2. Taylor expanded in y around inf 0.8

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

   3. Simplified 63.5

      \[\leadsto \color{blue}{\frac{y}{z} \cdot x} \]
      Proof
      (*.f64 (/.f64 y z) x): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 y x) z)): 45 points increase in error, 52 points decrease in error

   4. Applied egg-rr 0.8

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

   if -inf.0 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < -9.99999999999999954e-259 or 4.99999999982e-314 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < 1.99999999999999997e307

   1. Initial program 0.2

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

   if -9.99999999999999954e-259 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < 4.99999999982e-314

   1. Initial program 16.5

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

   2. Taylor expanded in z around inf 0.2

      \[\leadsto \color{blue}{\frac{x \cdot \left(y - -1 \cdot t\right)}{z} + \frac{t \cdot x}{{z}^{2}}} \]

   3. Simplified 0.2

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \left(\left(y + t\right) + \frac{t}{z}\right)} \]
      Proof
      (*.f64 (/.f64 x z) (+.f64 (+.f64 y t) (/.f64 t z))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 x z) (+.f64 (+.f64 y (Rewrite<= *-lft-identity_binary64 (*.f64 1 t))) (/.f64 t z))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 x z) (+.f64 (+.f64 y (*.f64 (Rewrite<= metadata-eval (neg.f64 -1)) t)) (/.f64 t z))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 x z) (+.f64 (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 y (*.f64 -1 t))) (/.f64 t z))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 (/.f64 x z) (-.f64 y (*.f64 -1 t))) (*.f64 (/.f64 x z) (/.f64 t z)))): 1 points increase in error, 1 points decrease in error
      (+.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 x (-.f64 y (*.f64 -1 t))) z)) (*.f64 (/.f64 x z) (/.f64 t z))): 33 points increase in error, 53 points decrease in error
      (+.f64 (/.f64 (*.f64 x (-.f64 y (*.f64 -1 t))) z) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 x t) (*.f64 z z)))): 20 points increase in error, 12 points decrease in error
      (+.f64 (/.f64 (*.f64 x (-.f64 y (*.f64 -1 t))) z) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 t x)) (*.f64 z z))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (*.f64 x (-.f64 y (*.f64 -1 t))) z) (/.f64 (*.f64 t x) (Rewrite<= unpow2_binary64 (pow.f64 z 2)))): 0 points increase in error, 0 points decrease in error

   4. Applied egg-rr 0.2

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \leq -\infty:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq -1 \cdot 10^{-258}:\\ \;\;\;\;\left(\frac{y}{z} - \frac{t}{1 - z}\right) \cdot x\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 5 \cdot 10^{-314}:\\ \;\;\;\;\frac{\left(y + t\right) + \frac{t}{z}}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\left(\frac{y}{z} - \frac{t}{1 - z}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]

Alternatives

Alternative 1
Error	22.5
Cost	848

\[\begin{array}{l} t_1 := x \cdot \frac{t}{z}\\ t_2 := \frac{x}{\frac{z}{y}}\\ \mathbf{if}\;t \leq -4.503870073152293 \cdot 10^{+116}:\\ \;\;\;\;\frac{x}{\frac{z}{t}}\\ \mathbf{elif}\;t \leq 2.4782270306691114 \cdot 10^{+36}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 8.060982234082767 \cdot 10^{+95}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 7.8 \cdot 10^{+180}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	22.4
Cost	848

\[\begin{array}{l} t_1 := x \cdot \frac{t}{z}\\ \mathbf{if}\;t \leq -4.503870073152293 \cdot 10^{+116}:\\ \;\;\;\;\frac{x}{\frac{z}{t}}\\ \mathbf{elif}\;t \leq 2.4782270306691114 \cdot 10^{+36}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;t \leq 8.060982234082767 \cdot 10^{+95}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 7.8 \cdot 10^{+180}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	22.5
Cost	848

\[\begin{array}{l} t_1 := x \cdot \frac{t}{z}\\ \mathbf{if}\;t \leq -4.503870073152293 \cdot 10^{+116}:\\ \;\;\;\;\frac{x}{\frac{z}{t}}\\ \mathbf{elif}\;t \leq 2.4782270306691114 \cdot 10^{+36}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;t \leq 8.060982234082767 \cdot 10^{+95}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 7.8 \cdot 10^{+180}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	22.5
Cost	848

\[\begin{array}{l} t_1 := x \cdot \frac{t}{z}\\ \mathbf{if}\;t \leq -4.503870073152293 \cdot 10^{+116}:\\ \;\;\;\;\frac{x}{\frac{z}{t}}\\ \mathbf{elif}\;t \leq 2.4782270306691114 \cdot 10^{+36}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;t \leq 8.060982234082767 \cdot 10^{+95}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 7.8 \cdot 10^{+180}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	15.5
Cost	712

\[\begin{array}{l} t_1 := \left(y + t\right) \cdot \frac{x}{z}\\ \mathbf{if}\;z \leq -4.4 \cdot 10^{-43}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.28 \cdot 10^{-17}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	9.5
Cost	712

\[\begin{array}{l} t_1 := \left(y + t\right) \cdot \frac{x}{z}\\ \mathbf{if}\;z \leq -1 \cdot 10^{+15}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 10^{-20}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	6.1
Cost	712

\[\begin{array}{l} t_1 := \frac{x}{\frac{z}{y + t}}\\ \mathbf{if}\;z \leq -1 \cdot 10^{+15}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-23}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	6.2
Cost	712

\[\begin{array}{l} t_1 := x \cdot \frac{y + t}{z}\\ \mathbf{if}\;z \leq -1 \cdot 10^{+15}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 10^{-35}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	33.6
Cost	584

\[\begin{array}{l} t_1 := x \cdot \frac{t}{z}\\ \mathbf{if}\;z \leq -2700000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-17}:\\ \;\;\;\;t \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	50.6
Cost	256

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

Reproduce

herbie shell --seed 2022308 
(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)))))