VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.6% → 96.2%

Time: 38.3s

Alternatives: 8

Speedup: 3.3×

• could not determine a ground truth

  1. f = 2.00000000000000007e154

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{4} \cdot f\\ t_1 := e^{t_0}\\ t_2 := e^{-t_0}\\ -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{t_1 + t_2}{t_1 - t_2}\right) \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* (/ PI 4.0) f)) (t_1 (exp t_0)) (t_2 (exp (- t_0))))
   (- (* (/ 1.0 (/ PI 4.0)) (log (/ (+ t_1 t_2) (- t_1 t_2)))))))

C

double code(double f) {
	double t_0 = (((double) M_PI) / 4.0) * f;
	double t_1 = exp(t_0);
	double t_2 = exp(-t_0);
	return -((1.0 / (((double) M_PI) / 4.0)) * log(((t_1 + t_2) / (t_1 - t_2))));
}

Java

public static double code(double f) {
	double t_0 = (Math.PI / 4.0) * f;
	double t_1 = Math.exp(t_0);
	double t_2 = Math.exp(-t_0);
	return -((1.0 / (Math.PI / 4.0)) * Math.log(((t_1 + t_2) / (t_1 - t_2))));
}

Python

def code(f):
	t_0 = (math.pi / 4.0) * f
	t_1 = math.exp(t_0)
	t_2 = math.exp(-t_0)
	return -((1.0 / (math.pi / 4.0)) * math.log(((t_1 + t_2) / (t_1 - t_2))))

Julia

function code(f)
	t_0 = Float64(Float64(pi / 4.0) * f)
	t_1 = exp(t_0)
	t_2 = exp(Float64(-t_0))
	return Float64(-Float64(Float64(1.0 / Float64(pi / 4.0)) * log(Float64(Float64(t_1 + t_2) / Float64(t_1 - t_2)))))
end

MATLAB

function tmp = code(f)
	t_0 = (pi / 4.0) * f;
	t_1 = exp(t_0);
	t_2 = exp(-t_0);
	tmp = -((1.0 / (pi / 4.0)) * log(((t_1 + t_2) / (t_1 - t_2))));
end

Wolfram

code[f_] := Block[{t$95$0 = N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision]}, Block[{t$95$1 = N[Exp[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Exp[(-t$95$0)], $MachinePrecision]}, (-N[(N[(1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision] * N[Log[N[(N[(t$95$1 + t$95$2), $MachinePrecision] / N[(t$95$1 - t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]]]

Initial Program: 6.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{4} \cdot f\\ t_1 := e^{t_0}\\ t_2 := e^{-t_0}\\ -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{t_1 + t_2}{t_1 - t_2}\right) \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* (/ PI 4.0) f)) (t_1 (exp t_0)) (t_2 (exp (- t_0))))
   (- (* (/ 1.0 (/ PI 4.0)) (log (/ (+ t_1 t_2) (- t_1 t_2)))))))

C

double code(double f) {
	double t_0 = (((double) M_PI) / 4.0) * f;
	double t_1 = exp(t_0);
	double t_2 = exp(-t_0);
	return -((1.0 / (((double) M_PI) / 4.0)) * log(((t_1 + t_2) / (t_1 - t_2))));
}

Java

public static double code(double f) {
	double t_0 = (Math.PI / 4.0) * f;
	double t_1 = Math.exp(t_0);
	double t_2 = Math.exp(-t_0);
	return -((1.0 / (Math.PI / 4.0)) * Math.log(((t_1 + t_2) / (t_1 - t_2))));
}

Python

def code(f):
	t_0 = (math.pi / 4.0) * f
	t_1 = math.exp(t_0)
	t_2 = math.exp(-t_0)
	return -((1.0 / (math.pi / 4.0)) * math.log(((t_1 + t_2) / (t_1 - t_2))))

Julia

function code(f)
	t_0 = Float64(Float64(pi / 4.0) * f)
	t_1 = exp(t_0)
	t_2 = exp(Float64(-t_0))
	return Float64(-Float64(Float64(1.0 / Float64(pi / 4.0)) * log(Float64(Float64(t_1 + t_2) / Float64(t_1 - t_2)))))
end

MATLAB

function tmp = code(f)
	t_0 = (pi / 4.0) * f;
	t_1 = exp(t_0);
	t_2 = exp(-t_0);
	tmp = -((1.0 / (pi / 4.0)) * log(((t_1 + t_2) / (t_1 - t_2))));
end

Wolfram

code[f_] := Block[{t$95$0 = N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision]}, Block[{t$95$1 = N[Exp[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Exp[(-t$95$0)], $MachinePrecision]}, (-N[(N[(1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision] * N[Log[N[(N[(t$95$1 + t$95$2), $MachinePrecision] / N[(t$95$1 - t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]]]

Alternative 1: 96.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ -4 \cdot \frac{\log \left(\frac{e^{-0.25 \cdot \left(f \cdot \pi\right)} + e^{\left(f \cdot \pi\right) \cdot 0.25}}{\mathsf{fma}\left(\pi, f \cdot 0.5, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, {\left(f \cdot \pi\right)}^{3} \cdot 0.005208333333333333\right)\right)}\right)}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  -4.0
  (/
   (log
    (/
     (+ (exp (* -0.25 (* f PI))) (exp (* (* f PI) 0.25)))
     (fma
      PI
      (* f 0.5)
      (fma
       (pow f 5.0)
       (* (pow PI 5.0) 1.6276041666666666e-5)
       (* (pow (* f PI) 3.0) 0.005208333333333333)))))
   PI)))

C

double code(double f) {
	return -4.0 * (log(((exp((-0.25 * (f * ((double) M_PI)))) + exp(((f * ((double) M_PI)) * 0.25))) / fma(((double) M_PI), (f * 0.5), fma(pow(f, 5.0), (pow(((double) M_PI), 5.0) * 1.6276041666666666e-5), (pow((f * ((double) M_PI)), 3.0) * 0.005208333333333333))))) / ((double) M_PI));
}

Julia

function code(f)
	return Float64(-4.0 * Float64(log(Float64(Float64(exp(Float64(-0.25 * Float64(f * pi))) + exp(Float64(Float64(f * pi) * 0.25))) / fma(pi, Float64(f * 0.5), fma((f ^ 5.0), Float64((pi ^ 5.0) * 1.6276041666666666e-5), Float64((Float64(f * pi) ^ 3.0) * 0.005208333333333333))))) / pi))
end

Wolfram

code[f_] := N[(-4.0 * N[(N[Log[N[(N[(N[Exp[N[(-0.25 * N[(f * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(N[(f * Pi), $MachinePrecision] * 0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(Pi * N[(f * 0.5), $MachinePrecision] + N[(N[Power[f, 5.0], $MachinePrecision] * N[(N[Power[Pi, 5.0], $MachinePrecision] * 1.6276041666666666e-5), $MachinePrecision] + N[(N[Power[N[(f * Pi), $MachinePrecision], 3.0], $MachinePrecision] * 0.005208333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

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Alternative 2: 96.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ -4 \cdot \frac{\log \left(\frac{e^{-0.25 \cdot \left(f \cdot \pi\right)} + e^{\left(f \cdot \pi\right) \cdot 0.25}}{\mathsf{fma}\left(\pi, f \cdot 0.5, {\left(f \cdot \pi\right)}^{3} \cdot 0.005208333333333333\right)}\right)}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  -4.0
  (/
   (log
    (/
     (+ (exp (* -0.25 (* f PI))) (exp (* (* f PI) 0.25)))
     (fma PI (* f 0.5) (* (pow (* f PI) 3.0) 0.005208333333333333))))
   PI)))

C

double code(double f) {
	return -4.0 * (log(((exp((-0.25 * (f * ((double) M_PI)))) + exp(((f * ((double) M_PI)) * 0.25))) / fma(((double) M_PI), (f * 0.5), (pow((f * ((double) M_PI)), 3.0) * 0.005208333333333333)))) / ((double) M_PI));
}

Julia

function code(f)
	return Float64(-4.0 * Float64(log(Float64(Float64(exp(Float64(-0.25 * Float64(f * pi))) + exp(Float64(Float64(f * pi) * 0.25))) / fma(pi, Float64(f * 0.5), Float64((Float64(f * pi) ^ 3.0) * 0.005208333333333333)))) / pi))
end

Wolfram

code[f_] := N[(-4.0 * N[(N[Log[N[(N[(N[Exp[N[(-0.25 * N[(f * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(N[(f * Pi), $MachinePrecision] * 0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(Pi * N[(f * 0.5), $MachinePrecision] + N[(N[Power[N[(f * Pi), $MachinePrecision], 3.0], $MachinePrecision] * 0.005208333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

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Alternative 3: 96.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(\log \left(\frac{4}{\pi}\right) + \left(0.5 \cdot \mathsf{fma}\left(f, 0, {f}^{2} \cdot \mathsf{fma}\left(0.5, \pi \cdot \left(\pi \cdot 0.08333333333333333\right), 0\right)\right) - \log f\right)\right) \cdot \frac{-4}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (+
   (log (/ 4.0 PI))
   (-
    (*
     0.5
     (fma
      f
      0.0
      (* (pow f 2.0) (fma 0.5 (* PI (* PI 0.08333333333333333)) 0.0))))
    (log f)))
  (/ -4.0 PI)))

C

double code(double f) {
	return (log((4.0 / ((double) M_PI))) + ((0.5 * fma(f, 0.0, (pow(f, 2.0) * fma(0.5, (((double) M_PI) * (((double) M_PI) * 0.08333333333333333)), 0.0)))) - log(f))) * (-4.0 / ((double) M_PI));
}

Julia

function code(f)
	return Float64(Float64(log(Float64(4.0 / pi)) + Float64(Float64(0.5 * fma(f, 0.0, Float64((f ^ 2.0) * fma(0.5, Float64(pi * Float64(pi * 0.08333333333333333)), 0.0)))) - log(f))) * Float64(-4.0 / pi))
end

Wolfram

code[f_] := N[(N[(N[Log[N[(4.0 / Pi), $MachinePrecision]], $MachinePrecision] + N[(N[(0.5 * N[(f * 0.0 + N[(N[Power[f, 2.0], $MachinePrecision] * N[(0.5 * N[(Pi * N[(Pi * 0.08333333333333333), $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Log[f], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-4.0 / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

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Alternative 4: 95.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ -4 \cdot \frac{\log \left(\frac{e^{-0.25 \cdot \left(f \cdot \pi\right)} + e^{\left(f \cdot \pi\right) \cdot 0.25}}{\pi \cdot \left(f \cdot 0.5\right)}\right)}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  -4.0
  (/
   (log
    (/ (+ (exp (* -0.25 (* f PI))) (exp (* (* f PI) 0.25))) (* PI (* f 0.5))))
   PI)))

C

double code(double f) {
	return -4.0 * (log(((exp((-0.25 * (f * ((double) M_PI)))) + exp(((f * ((double) M_PI)) * 0.25))) / (((double) M_PI) * (f * 0.5)))) / ((double) M_PI));
}

Java

public static double code(double f) {
	return -4.0 * (Math.log(((Math.exp((-0.25 * (f * Math.PI))) + Math.exp(((f * Math.PI) * 0.25))) / (Math.PI * (f * 0.5)))) / Math.PI);
}

Python

def code(f):
	return -4.0 * (math.log(((math.exp((-0.25 * (f * math.pi))) + math.exp(((f * math.pi) * 0.25))) / (math.pi * (f * 0.5)))) / math.pi)

Julia

function code(f)
	return Float64(-4.0 * Float64(log(Float64(Float64(exp(Float64(-0.25 * Float64(f * pi))) + exp(Float64(Float64(f * pi) * 0.25))) / Float64(pi * Float64(f * 0.5)))) / pi))
end

MATLAB

function tmp = code(f)
	tmp = -4.0 * (log(((exp((-0.25 * (f * pi))) + exp(((f * pi) * 0.25))) / (pi * (f * 0.5)))) / pi);
end

Wolfram

code[f_] := N[(-4.0 * N[(N[Log[N[(N[(N[Exp[N[(-0.25 * N[(f * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(N[(f * Pi), $MachinePrecision] * 0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(Pi * N[(f * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

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Alternative 5: 95.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ -4 \cdot \frac{\log \left(\frac{4}{\pi}\right) + \log \left(\frac{1}{f}\right)}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (* -4.0 (/ (+ (log (/ 4.0 PI)) (log (/ 1.0 f))) PI)))

C

double code(double f) {
	return -4.0 * ((log((4.0 / ((double) M_PI))) + log((1.0 / f))) / ((double) M_PI));
}

Java

public static double code(double f) {
	return -4.0 * ((Math.log((4.0 / Math.PI)) + Math.log((1.0 / f))) / Math.PI);
}

Python

def code(f):
	return -4.0 * ((math.log((4.0 / math.pi)) + math.log((1.0 / f))) / math.pi)

Julia

function code(f)
	return Float64(-4.0 * Float64(Float64(log(Float64(4.0 / pi)) + log(Float64(1.0 / f))) / pi))
end

MATLAB

function tmp = code(f)
	tmp = -4.0 * ((log((4.0 / pi)) + log((1.0 / f))) / pi);
end

Wolfram

code[f_] := N[(-4.0 * N[(N[(N[Log[N[(4.0 / Pi), $MachinePrecision]], $MachinePrecision] + N[Log[N[(1.0 / f), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

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Alternative 6: 95.5% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \frac{-4}{\pi} \cdot \log \left(\frac{4}{f \cdot \pi}\right) \end{array} \]

FPCore

(FPCore (f) :precision binary64 (* (/ -4.0 PI) (log (/ 4.0 (* f PI)))))

C

double code(double f) {
	return (-4.0 / ((double) M_PI)) * log((4.0 / (f * ((double) M_PI))));
}

Java

public static double code(double f) {
	return (-4.0 / Math.PI) * Math.log((4.0 / (f * Math.PI)));
}

Python

def code(f):
	return (-4.0 / math.pi) * math.log((4.0 / (f * math.pi)))

Julia

function code(f)
	return Float64(Float64(-4.0 / pi) * log(Float64(4.0 / Float64(f * pi))))
end

MATLAB

function tmp = code(f)
	tmp = (-4.0 / pi) * log((4.0 / (f * pi)));
end

Wolfram

code[f_] := N[(N[(-4.0 / Pi), $MachinePrecision] * N[Log[N[(4.0 / N[(f * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

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Alternative 7: 95.6% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \frac{-4 \cdot \log \left(\frac{4}{f \cdot \pi}\right)}{\pi} \end{array} \]

FPCore

(FPCore (f) :precision binary64 (/ (* -4.0 (log (/ 4.0 (* f PI)))) PI))

C

double code(double f) {
	return (-4.0 * log((4.0 / (f * ((double) M_PI))))) / ((double) M_PI);
}

Java

public static double code(double f) {
	return (-4.0 * Math.log((4.0 / (f * Math.PI)))) / Math.PI;
}

Python

def code(f):
	return (-4.0 * math.log((4.0 / (f * math.pi)))) / math.pi

Julia

function code(f)
	return Float64(Float64(-4.0 * log(Float64(4.0 / Float64(f * pi)))) / pi)
end

MATLAB

function tmp = code(f)
	tmp = (-4.0 * log((4.0 / (f * pi)))) / pi;
end

Wolfram

code[f_] := N[(N[(-4.0 * N[Log[N[(4.0 / N[(f * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]

Derivation

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Alternative 8: 3.1% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ -4 \cdot \frac{-\log 0}{\pi} \end{array} \]

FPCore

(FPCore (f) :precision binary64 (* -4.0 (/ (- (log 0.0)) PI)))

C

double code(double f) {
	return -4.0 * (-log(0.0) / ((double) M_PI));
}

Java

public static double code(double f) {
	return -4.0 * (-Math.log(0.0) / Math.PI);
}

Python

def code(f):
	return -4.0 * (-math.log(0.0) / math.pi)

Julia

function code(f)
	return Float64(-4.0 * Float64(Float64(-log(0.0)) / pi))
end

MATLAB

function tmp = code(f)
	tmp = -4.0 * (-log(0.0) / pi);
end

Wolfram

code[f_] := N[(-4.0 * N[((-N[Log[0.0], $MachinePrecision]) / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

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Reproduce

herbie shell --seed 2023350 
(FPCore (f)
  :name "VandenBroeck and Keller, Equation (20)"
  :precision binary64
  (- (* (/ 1.0 (/ PI 4.0)) (log (/ (+ (exp (* (/ PI 4.0) f)) (exp (- (* (/ PI 4.0) f)))) (- (exp (* (/ PI 4.0) f)) (exp (- (* (/ PI 4.0) f)))))))))