System.Random.MWC.Distributions:truncatedExp from mwc-random-0.13.3.2

Percentage Accurate: 61.8% → 92.2%

Time: 17.5s

Alternatives: 11

Speedup: 11.3×

• could not determine a ground truth

  1. x = 2.6789304320582704e-120
  2. y = 4.4641169147666116e+88
  3. z = 1.986112752845242e+256
  4. t = 2.6676981921529202e-166

Specification

Math

\[\begin{array}{l} \\ x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- x (/ (log (+ (- 1.0 y) (* y (exp z)))) t)))

C

double code(double x, double y, double z, double t) {
	return x - (log(((1.0 - y) + (y * exp(z)))) / t);
}

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 - (log(((1.0d0 - y) + (y * exp(z)))) / t)
end function

Java

public static double code(double x, double y, double z, double t) {
	return x - (Math.log(((1.0 - y) + (y * Math.exp(z)))) / t);
}

Python

def code(x, y, z, t):
	return x - (math.log(((1.0 - y) + (y * math.exp(z)))) / t)

Julia

function code(x, y, z, t)
	return Float64(x - Float64(log(Float64(Float64(1.0 - y) + Float64(y * exp(z)))) / t))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x - (log(((1.0 - y) + (y * exp(z)))) / t);
end

Wolfram

code[x_, y_, z_, t_] := N[(x - N[(N[Log[N[(N[(1.0 - y), $MachinePrecision] + N[(y * N[Exp[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]

Initial Program: 61.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- x (/ (log (+ (- 1.0 y) (* y (exp z)))) t)))

C

double code(double x, double y, double z, double t) {
	return x - (log(((1.0 - y) + (y * exp(z)))) / t);
}

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 - (log(((1.0d0 - y) + (y * exp(z)))) / t)
end function

Java

public static double code(double x, double y, double z, double t) {
	return x - (Math.log(((1.0 - y) + (y * Math.exp(z)))) / t);
}

Python

def code(x, y, z, t):
	return x - (math.log(((1.0 - y) + (y * math.exp(z)))) / t)

Julia

function code(x, y, z, t)
	return Float64(x - Float64(log(Float64(Float64(1.0 - y) + Float64(y * exp(z)))) / t))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x - (log(((1.0 - y) + (y * exp(z)))) / t);
end

Wolfram

code[x_, y_, z_, t_] := N[(x - N[(N[Log[N[(N[(1.0 - y), $MachinePrecision] + N[(y * N[Exp[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]

Alternative 1: 92.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{t}, \mathsf{log1p}\left(\mathsf{fma}\left(e^{z}, y, -y\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (exp z) 0.5)
   (fma (/ -1.0 t) (log1p (fma (exp z) y (- y))) x)
   (- x (/ (* y z) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (exp(z) <= 0.5) {
		tmp = fma((-1.0 / t), log1p(fma(exp(z), y, -y)), x);
	} else {
		tmp = x - ((y * z) / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (exp(z) <= 0.5)
		tmp = fma(Float64(-1.0 / t), log1p(fma(exp(z), y, Float64(-y))), x);
	else
		tmp = Float64(x - Float64(Float64(y * z) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.5], N[(N[(-1.0 / t), $MachinePrecision] * N[Log[1 + N[(N[Exp[z], $MachinePrecision] * y + (-y)), $MachinePrecision]], $MachinePrecision] + x), $MachinePrecision], N[(x - N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.5

   1. Initial program 84.5%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. div-inv N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{\mathsf{neg}\left(t\right)}, \log \left(\left(1 - y\right) + y \cdot e^{z}\right), x\right)} \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{t}, \mathsf{log1p}\left(\mathsf{fma}\left(e^{z}, y, -y\right)\right), x\right)} \]

   if 0.5 < (exp.f64 z)

   1. Initial program 56.5%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto x - \frac{\color{blue}{y \cdot z}}{t} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 93.8

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

   5. Applied rewrites 93.8%

      \[\leadsto x - \frac{\color{blue}{z \cdot y}}{t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{t}, \mathsf{log1p}\left(\mathsf{fma}\left(e^{z}, y, -y\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot z}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{expm1}\left(z\right) \cdot y\\ \mathbf{if}\;y \cdot e^{z} + \left(1 - y\right) \leq 2:\\ \;\;\;\;x - \frac{t\_1}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log t\_1}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (expm1 z) y)))
   (if (<= (+ (* y (exp z)) (- 1.0 y)) 2.0)
     (- x (/ t_1 t))
     (- x (/ (log t_1) t)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = expm1(z) * y;
	double tmp;
	if (((y * exp(z)) + (1.0 - y)) <= 2.0) {
		tmp = x - (t_1 / t);
	} else {
		tmp = x - (log(t_1) / t);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.expm1(z) * y;
	double tmp;
	if (((y * Math.exp(z)) + (1.0 - y)) <= 2.0) {
		tmp = x - (t_1 / t);
	} else {
		tmp = x - (Math.log(t_1) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.expm1(z) * y
	tmp = 0
	if ((y * math.exp(z)) + (1.0 - y)) <= 2.0:
		tmp = x - (t_1 / t)
	else:
		tmp = x - (math.log(t_1) / t)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(expm1(z) * y)
	tmp = 0.0
	if (Float64(Float64(y * exp(z)) + Float64(1.0 - y)) <= 2.0)
		tmp = Float64(x - Float64(t_1 / t));
	else
		tmp = Float64(x - Float64(log(t_1) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(Exp[z] - 1), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[N[(N[(y * N[Exp[z], $MachinePrecision]), $MachinePrecision] + N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 2.0], N[(x - N[(t$95$1 / t), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Log[t$95$1], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z))) < 2

   1. Initial program 61.3%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto x - \frac{\color{blue}{y \cdot \left(e^{z} - 1\right)}}{t} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 94.9

         \[\leadsto x - \frac{\color{blue}{\mathsf{expm1}\left(z\right)} \cdot y}{t} \]

   5. Applied rewrites 94.9%

      \[\leadsto x - \frac{\color{blue}{\mathsf{expm1}\left(z\right) \cdot y}}{t} \]

   if 2 < (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z)))

   1. Initial program 95.8%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around -inf

      \[\leadsto x - \frac{\log \color{blue}{\left(-1 \cdot \left(y \cdot \left(1 + -1 \cdot e^{z}\right)\right)\right)}}{t} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

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

      2. neg-mul-1 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. distribute-lft-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. distribute-rgt-neg-out N/A

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

      8. remove-double-neg N/A

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

      9. cancel-sign-sub-inv N/A

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

      10. distribute-lft-out-- N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-expm1.f64 95.8

          \[\leadsto x - \frac{\log \left(\color{blue}{\mathsf{expm1}\left(z\right)} \cdot y\right)}{t} \]

   5. Applied rewrites 95.8%

      \[\leadsto x - \frac{\log \color{blue}{\left(\mathsf{expm1}\left(z\right) \cdot y\right)}}{t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot e^{z} + \left(1 - y\right) \leq 2:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right) \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{expm1}\left(z\right) \cdot y\right)}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot e^{z} + \left(1 - y\right) \leq 4 \cdot 10^{+91}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right) \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log 1}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ (* y (exp z)) (- 1.0 y)) 4e+91)
   (- x (/ (* (expm1 z) y) t))
   (- x (/ (log 1.0) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((y * exp(z)) + (1.0 - y)) <= 4e+91) {
		tmp = x - ((expm1(z) * y) / t);
	} else {
		tmp = x - (log(1.0) / t);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((y * Math.exp(z)) + (1.0 - y)) <= 4e+91) {
		tmp = x - ((Math.expm1(z) * y) / t);
	} else {
		tmp = x - (Math.log(1.0) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((y * math.exp(z)) + (1.0 - y)) <= 4e+91:
		tmp = x - ((math.expm1(z) * y) / t)
	else:
		tmp = x - (math.log(1.0) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(y * exp(z)) + Float64(1.0 - y)) <= 4e+91)
		tmp = Float64(x - Float64(Float64(expm1(z) * y) / t));
	else
		tmp = Float64(x - Float64(log(1.0) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(y * N[Exp[z], $MachinePrecision]), $MachinePrecision] + N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 4e+91], N[(x - N[(N[(N[(Exp[z] - 1), $MachinePrecision] * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Log[1.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z))) < 4.00000000000000032e91

   1. Initial program 62.6%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto x - \frac{\color{blue}{y \cdot \left(e^{z} - 1\right)}}{t} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 93.6

         \[\leadsto x - \frac{\color{blue}{\mathsf{expm1}\left(z\right)} \cdot y}{t} \]

   5. Applied rewrites 93.6%

      \[\leadsto x - \frac{\color{blue}{\mathsf{expm1}\left(z\right) \cdot y}}{t} \]

   if 4.00000000000000032e91 < (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z)))

   1. Initial program 93.7%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto x - \frac{\log \color{blue}{1}}{t} \]
   4. Step-by-step derivation

   5. Applied rewrites 55.3%

      \[\leadsto x - \frac{\log \color{blue}{1}}{t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot e^{z} + \left(1 - y\right) \leq 4 \cdot 10^{+91}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right) \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log 1}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.55 \cdot 10^{+266}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z, 0.5\right) \cdot y, z, y\right), z, 1\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right) \cdot y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -2.55e+266)
   (-
    x
    (/ (log (fma (fma (* (fma 0.16666666666666666 z 0.5) y) z y) z 1.0)) t))
   (- x (/ (* (expm1 z) y) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -2.55e+266) {
		tmp = x - (log(fma(fma((fma(0.16666666666666666, z, 0.5) * y), z, y), z, 1.0)) / t);
	} else {
		tmp = x - ((expm1(z) * y) / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -2.55e+266)
		tmp = Float64(x - Float64(log(fma(fma(Float64(fma(0.16666666666666666, z, 0.5) * y), z, y), z, 1.0)) / t));
	else
		tmp = Float64(x - Float64(Float64(expm1(z) * y) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -2.55e+266], N[(x - N[(N[Log[N[(N[(N[(N[(0.16666666666666666 * z + 0.5), $MachinePrecision] * y), $MachinePrecision] * z + y), $MachinePrecision] * z + 1.0), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(N[(Exp[z] - 1), $MachinePrecision] * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.55000000000000004e266

   1. Initial program 23.8%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto x - \frac{\log \color{blue}{\left(1 + z \cdot \left(y + z \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y\right)\right)\right)}}{t} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x - \frac{\log \color{blue}{\left(z \cdot \left(y + z \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y\right)\right) + 1\right)}}{t} \]

      2. *-commutative N/A

         \[\leadsto x - \frac{\log \left(\color{blue}{\left(y + z \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y\right)\right) \cdot z} + 1\right)}{t} \]

      3. lower-fma.f64 N/A

         \[\leadsto x - \frac{\log \color{blue}{\left(\mathsf{fma}\left(y + z \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y\right), z, 1\right)\right)}}{t} \]

      4. +-commutative N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\color{blue}{z \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y\right) + y}, z, 1\right)\right)}{t} \]

      5. *-commutative N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y\right) \cdot z} + y, z, 1\right)\right)}{t} \]

      6. lower-fma.f64 N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y, z, y\right)}, z, 1\right)\right)}{t} \]

      7. *-commutative N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6} \cdot \color{blue}{\left(z \cdot y\right)} + \frac{1}{2} \cdot y, z, y\right), z, 1\right)\right)}{t} \]

      8. associate-*r* N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{6} \cdot z\right) \cdot y} + \frac{1}{2} \cdot y, z, y\right), z, 1\right)\right)}{t} \]

      9. distribute-rgt-out N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{y \cdot \left(\frac{1}{6} \cdot z + \frac{1}{2}\right)}, z, y\right), z, 1\right)\right)}{t} \]

      10. lower-*.f64 N/A

          \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{y \cdot \left(\frac{1}{6} \cdot z + \frac{1}{2}\right)}, z, y\right), z, 1\right)\right)}{t} \]

      11. lower-fma.f64 81.9

          \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot \color{blue}{\mathsf{fma}\left(0.16666666666666666, z, 0.5\right)}, z, y\right), z, 1\right)\right)}{t} \]

   5. Applied rewrites 81.9%

      \[\leadsto x - \frac{\log \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot \mathsf{fma}\left(0.16666666666666666, z, 0.5\right), z, y\right), z, 1\right)\right)}}{t} \]

   if -2.55000000000000004e266 < y

   1. Initial program 66.1%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto x - \frac{\color{blue}{y \cdot \left(e^{z} - 1\right)}}{t} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 91.8

         \[\leadsto x - \frac{\color{blue}{\mathsf{expm1}\left(z\right)} \cdot y}{t} \]

   5. Applied rewrites 91.8%

      \[\leadsto x - \frac{\color{blue}{\mathsf{expm1}\left(z\right) \cdot y}}{t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.55 \cdot 10^{+266}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z, 0.5\right) \cdot y, z, y\right), z, 1\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right) \cdot y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.7 \cdot 10^{+265}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(z, y, 1\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right) \cdot y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -5.7e+265)
   (- x (/ (log (fma z y 1.0)) t))
   (- x (/ (* (expm1 z) y) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -5.7e+265) {
		tmp = x - (log(fma(z, y, 1.0)) / t);
	} else {
		tmp = x - ((expm1(z) * y) / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -5.7e+265)
		tmp = Float64(x - Float64(log(fma(z, y, 1.0)) / t));
	else
		tmp = Float64(x - Float64(Float64(expm1(z) * y) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -5.7e+265], N[(x - N[(N[Log[N[(z * y + 1.0), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(N[(Exp[z] - 1), $MachinePrecision] * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.6999999999999999e265

   1. Initial program 23.8%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto x - \frac{\log \color{blue}{\left(1 + y \cdot z\right)}}{t} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 81.5

         \[\leadsto x - \frac{\log \color{blue}{\left(\mathsf{fma}\left(z, y, 1\right)\right)}}{t} \]

   5. Applied rewrites 81.5%

      \[\leadsto x - \frac{\log \color{blue}{\left(\mathsf{fma}\left(z, y, 1\right)\right)}}{t} \]

   if -5.6999999999999999e265 < y

   1. Initial program 66.1%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto x - \frac{\color{blue}{y \cdot \left(e^{z} - 1\right)}}{t} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 91.8

         \[\leadsto x - \frac{\color{blue}{\mathsf{expm1}\left(z\right)} \cdot y}{t} \]

   5. Applied rewrites 91.8%

      \[\leadsto x - \frac{\color{blue}{\mathsf{expm1}\left(z\right) \cdot y}}{t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 87.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3 \cdot 10^{+48}:\\ \;\;\;\;x - \frac{\log 1}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{t} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -3e+48) (- x (/ (log 1.0) t)) (- x (* (/ (expm1 z) t) y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -3e+48) {
		tmp = x - (log(1.0) / t);
	} else {
		tmp = x - ((expm1(z) / t) * y);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -3e+48) {
		tmp = x - (Math.log(1.0) / t);
	} else {
		tmp = x - ((Math.expm1(z) / t) * y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -3e+48:
		tmp = x - (math.log(1.0) / t)
	else:
		tmp = x - ((math.expm1(z) / t) * y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -3e+48)
		tmp = Float64(x - Float64(log(1.0) / t));
	else
		tmp = Float64(x - Float64(Float64(expm1(z) / t) * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -3e+48], N[(x - N[(N[Log[1.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3e48

   1. Initial program 79.3%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto x - \frac{\log \color{blue}{1}}{t} \]
   4. Step-by-step derivation

   5. Applied rewrites 91.5%

      \[\leadsto x - \frac{\log \color{blue}{1}}{t} \]

   if -3e48 < t

   1. Initial program 60.0%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto x - \color{blue}{\frac{y \cdot \left(e^{z} - 1\right)}{t}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. div-sub N/A

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

      6. lower-/.f64 N/A

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

      7. lower-expm1.f64 90.4

         \[\leadsto x - \frac{\color{blue}{\mathsf{expm1}\left(z\right)}}{t} \cdot y \]

   5. Applied rewrites 90.4%

      \[\leadsto x - \color{blue}{\frac{\mathsf{expm1}\left(z\right)}{t} \cdot y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 81.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{+36}:\\ \;\;\;\;x - \frac{\log 1}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -7.8e+36) (- x (/ (log 1.0) t)) (- x (/ (* y z) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -7.8e+36) {
		tmp = x - (log(1.0) / t);
	} else {
		tmp = x - ((y * z) / t);
	}
	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
    real(8) :: tmp
    if (z <= (-7.8d+36)) then
        tmp = x - (log(1.0d0) / t)
    else
        tmp = x - ((y * z) / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -7.8e+36) {
		tmp = x - (Math.log(1.0) / t);
	} else {
		tmp = x - ((y * z) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -7.8e+36:
		tmp = x - (math.log(1.0) / t)
	else:
		tmp = x - ((y * z) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -7.8e+36)
		tmp = Float64(x - Float64(log(1.0) / t));
	else
		tmp = Float64(x - Float64(Float64(y * z) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -7.8e+36)
		tmp = x - (log(1.0) / t);
	else
		tmp = x - ((y * z) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -7.8e+36], N[(x - N[(N[Log[1.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -7.80000000000000042e36

   1. Initial program 84.7%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto x - \frac{\log \color{blue}{1}}{t} \]
   4. Step-by-step derivation

   5. Applied rewrites 69.0%

      \[\leadsto x - \frac{\log \color{blue}{1}}{t} \]

   if -7.80000000000000042e36 < z

   1. Initial program 58.2%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto x - \frac{\color{blue}{y \cdot z}}{t} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 92.2

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

   5. Applied rewrites 92.2%

      \[\leadsto x - \frac{\color{blue}{z \cdot y}}{t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{+36}:\\ \;\;\;\;x - \frac{\log 1}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot z}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.8% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ x - \frac{y \cdot z}{t} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- x (/ (* y z) t)))

C

double code(double x, double y, double z, double t) {
	return x - ((y * z) / t);
}

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)
end function

Java

public static double code(double x, double y, double z, double t) {
	return x - ((y * z) / t);
}

Python

def code(x, y, z, t):
	return x - ((y * z) / t)

Julia

function code(x, y, z, t)
	return Float64(x - Float64(Float64(y * z) / t))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x - ((y * z) / t);
end

Wolfram

code[x_, y_, z_, t_] := N[(x - N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.6%

   \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
2. Add Preprocessing

3. Taylor expanded in z around 0

   \[\leadsto x - \frac{\color{blue}{y \cdot z}}{t} \]
4. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f64 80.3

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

5. Applied rewrites 80.3%

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

6. Final simplification 80.3%

   \[\leadsto x - \frac{y \cdot z}{t} \]
7. Add Preprocessing

Alternative 9: 75.5% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ x - \frac{z}{t} \cdot y \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- x (* (/ z t) y)))

C

double code(double x, double y, double z, double t) {
	return x - ((z / t) * y);
}

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 - ((z / t) * y)
end function

Java

public static double code(double x, double y, double z, double t) {
	return x - ((z / t) * y);
}

Python

def code(x, y, z, t):
	return x - ((z / t) * y)

Julia

function code(x, y, z, t)
	return Float64(x - Float64(Float64(z / t) * y))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x - ((z / t) * y);
end

Wolfram

code[x_, y_, z_, t_] := N[(x - N[(N[(z / t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.6%

   \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
2. Add Preprocessing

3. Taylor expanded in z around 0

   \[\leadsto x - \color{blue}{\frac{y \cdot z}{t}} \]
4. Step-by-step derivation

   1. associate-*l/ N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 77.7

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

5. Applied rewrites 77.7%

   \[\leadsto x - \color{blue}{\frac{y}{t} \cdot z} \]
6. Step-by-step derivation

7. Applied rewrites 79.0%

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

8. Final simplification 79.0%

   \[\leadsto x - \frac{z}{t} \cdot y \]
9. Add Preprocessing

Alternative 10: 14.1% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-z\right) \cdot y}{t} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (/ (* (- z) y) t))

C

double code(double x, double y, double z, double t) {
	return (-z * y) / t;
}

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 = (-z * y) / t
end function

Java

public static double code(double x, double y, double z, double t) {
	return (-z * y) / t;
}

Python

def code(x, y, z, t):
	return (-z * y) / t

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(-z) * y) / t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (-z * y) / t;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[((-z) * y), $MachinePrecision] / t), $MachinePrecision]

Derivation

1. Initial program 64.6%

   \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{\log \left(\left(1 + y \cdot e^{z}\right) - y\right)}{t}} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\log \left(\left(1 + y \cdot e^{z}\right) - y\right)}{t}\right)} \]

   2. distribute-neg-frac2 N/A

      \[\leadsto \color{blue}{\frac{\log \left(\left(1 + y \cdot e^{z}\right) - y\right)}{\mathsf{neg}\left(t\right)}} \]

   3. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\log \left(\left(1 + y \cdot e^{z}\right) - y\right)}{\mathsf{neg}\left(t\right)}} \]

   4. sub-neg N/A

      \[\leadsto \frac{\log \color{blue}{\left(\left(1 + y \cdot e^{z}\right) + \left(\mathsf{neg}\left(y\right)\right)\right)}}{\mathsf{neg}\left(t\right)} \]

   5. associate-+l+ N/A

      \[\leadsto \frac{\log \color{blue}{\left(1 + \left(y \cdot e^{z} + \left(\mathsf{neg}\left(y\right)\right)\right)\right)}}{\mathsf{neg}\left(t\right)} \]

   6. sub-neg N/A

      \[\leadsto \frac{\log \left(1 + \color{blue}{\left(y \cdot e^{z} - y\right)}\right)}{\mathsf{neg}\left(t\right)} \]

   7. *-rgt-identity N/A

      \[\leadsto \frac{\log \left(1 + \left(y \cdot e^{z} - \color{blue}{y \cdot 1}\right)\right)}{\mathsf{neg}\left(t\right)} \]

   8. distribute-lft-out-- N/A

      \[\leadsto \frac{\log \left(1 + \color{blue}{y \cdot \left(e^{z} - 1\right)}\right)}{\mathsf{neg}\left(t\right)} \]

   9. lower-log1p.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(y \cdot \left(e^{z} - 1\right)\right)}}{\mathsf{neg}\left(t\right)} \]

   10. *-commutative N/A

       \[\leadsto \frac{\mathsf{log1p}\left(\color{blue}{\left(e^{z} - 1\right) \cdot y}\right)}{\mathsf{neg}\left(t\right)} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{log1p}\left(\color{blue}{\left(e^{z} - 1\right) \cdot y}\right)}{\mathsf{neg}\left(t\right)} \]

   12. lower-expm1.f64 N/A

       \[\leadsto \frac{\mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(z\right)} \cdot y\right)}{\mathsf{neg}\left(t\right)} \]

   13. lower-neg.f64 25.4

       \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(z\right) \cdot y\right)}{\color{blue}{-t}} \]

5. Applied rewrites 25.4%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(\mathsf{expm1}\left(z\right) \cdot y\right)}{-t}} \]

6. Taylor expanded in z around 0

   \[\leadsto -1 \cdot \color{blue}{\frac{y \cdot z}{t}} \]
7. Step-by-step derivation

8. Applied rewrites 13.6%

   \[\leadsto \left(-z\right) \cdot \color{blue}{\frac{y}{t}} \]
9. Step-by-step derivation

10. Applied rewrites 14.9%

    \[\leadsto \frac{\left(-z\right) \cdot y}{t} \]
11. Add Preprocessing

Alternative 11: 14.7% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ \frac{-z}{t} \cdot y \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* (/ (- z) t) y))

C

double code(double x, double y, double z, double t) {
	return (-z / t) * y;
}

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 = (-z / t) * y
end function

Java

public static double code(double x, double y, double z, double t) {
	return (-z / t) * y;
}

Python

def code(x, y, z, t):
	return (-z / t) * y

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(-z) / t) * y)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (-z / t) * y;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[((-z) / t), $MachinePrecision] * y), $MachinePrecision]

Derivation

1. Initial program 64.6%

   \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{\log \left(\left(1 + y \cdot e^{z}\right) - y\right)}{t}} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\log \left(\left(1 + y \cdot e^{z}\right) - y\right)}{t}\right)} \]

   2. distribute-neg-frac2 N/A

      \[\leadsto \color{blue}{\frac{\log \left(\left(1 + y \cdot e^{z}\right) - y\right)}{\mathsf{neg}\left(t\right)}} \]

   3. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\log \left(\left(1 + y \cdot e^{z}\right) - y\right)}{\mathsf{neg}\left(t\right)}} \]

   4. sub-neg N/A

      \[\leadsto \frac{\log \color{blue}{\left(\left(1 + y \cdot e^{z}\right) + \left(\mathsf{neg}\left(y\right)\right)\right)}}{\mathsf{neg}\left(t\right)} \]

   5. associate-+l+ N/A

      \[\leadsto \frac{\log \color{blue}{\left(1 + \left(y \cdot e^{z} + \left(\mathsf{neg}\left(y\right)\right)\right)\right)}}{\mathsf{neg}\left(t\right)} \]

   6. sub-neg N/A

      \[\leadsto \frac{\log \left(1 + \color{blue}{\left(y \cdot e^{z} - y\right)}\right)}{\mathsf{neg}\left(t\right)} \]

   7. *-rgt-identity N/A

      \[\leadsto \frac{\log \left(1 + \left(y \cdot e^{z} - \color{blue}{y \cdot 1}\right)\right)}{\mathsf{neg}\left(t\right)} \]

   8. distribute-lft-out-- N/A

      \[\leadsto \frac{\log \left(1 + \color{blue}{y \cdot \left(e^{z} - 1\right)}\right)}{\mathsf{neg}\left(t\right)} \]

   9. lower-log1p.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(y \cdot \left(e^{z} - 1\right)\right)}}{\mathsf{neg}\left(t\right)} \]

   10. *-commutative N/A

       \[\leadsto \frac{\mathsf{log1p}\left(\color{blue}{\left(e^{z} - 1\right) \cdot y}\right)}{\mathsf{neg}\left(t\right)} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{log1p}\left(\color{blue}{\left(e^{z} - 1\right) \cdot y}\right)}{\mathsf{neg}\left(t\right)} \]

   12. lower-expm1.f64 N/A

       \[\leadsto \frac{\mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(z\right)} \cdot y\right)}{\mathsf{neg}\left(t\right)} \]

   13. lower-neg.f64 25.4

       \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(z\right) \cdot y\right)}{\color{blue}{-t}} \]

5. Applied rewrites 25.4%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(\mathsf{expm1}\left(z\right) \cdot y\right)}{-t}} \]

6. Taylor expanded in z around 0

   \[\leadsto -1 \cdot \color{blue}{\frac{y \cdot z}{t}} \]
7. Step-by-step derivation

8. Applied rewrites 13.6%

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

9. Taylor expanded in y around 0

   \[\leadsto -1 \cdot \frac{y \cdot z}{\color{blue}{t}} \]
10. Step-by-step derivation

11. Applied rewrites 14.1%

    \[\leadsto \frac{-z}{t} \cdot y \]
12. Add Preprocessing

Developer Target 1: 75.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-0.5}{y \cdot t}\\ \mathbf{if}\;z < -2.8874623088207947 \cdot 10^{+119}:\\ \;\;\;\;\left(x - \frac{t\_1}{z \cdot z}\right) - t\_1 \cdot \frac{\frac{2}{z}}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + z \cdot y\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- 0.5) (* y t))))
   (if (< z -2.8874623088207947e+119)
     (- (- x (/ t_1 (* z z))) (* t_1 (/ (/ 2.0 z) (* z z))))
     (- x (/ (log (+ 1.0 (* z y))) t)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = -0.5 / (y * t);
	double tmp;
	if (z < -2.8874623088207947e+119) {
		tmp = (x - (t_1 / (z * z))) - (t_1 * ((2.0 / z) / (z * z)));
	} else {
		tmp = x - (log((1.0 + (z * y))) / t);
	}
	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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -0.5d0 / (y * t)
    if (z < (-2.8874623088207947d+119)) then
        tmp = (x - (t_1 / (z * z))) - (t_1 * ((2.0d0 / z) / (z * z)))
    else
        tmp = x - (log((1.0d0 + (z * y))) / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = -0.5 / (y * t);
	double tmp;
	if (z < -2.8874623088207947e+119) {
		tmp = (x - (t_1 / (z * z))) - (t_1 * ((2.0 / z) / (z * z)));
	} else {
		tmp = x - (Math.log((1.0 + (z * y))) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = -0.5 / (y * t)
	tmp = 0
	if z < -2.8874623088207947e+119:
		tmp = (x - (t_1 / (z * z))) - (t_1 * ((2.0 / z) / (z * z)))
	else:
		tmp = x - (math.log((1.0 + (z * y))) / t)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(-0.5) / Float64(y * t))
	tmp = 0.0
	if (z < -2.8874623088207947e+119)
		tmp = Float64(Float64(x - Float64(t_1 / Float64(z * z))) - Float64(t_1 * Float64(Float64(2.0 / z) / Float64(z * z))));
	else
		tmp = Float64(x - Float64(log(Float64(1.0 + Float64(z * y))) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = -0.5 / (y * t);
	tmp = 0.0;
	if (z < -2.8874623088207947e+119)
		tmp = (x - (t_1 / (z * z))) - (t_1 * ((2.0 / z) / (z * z)));
	else
		tmp = x - (log((1.0 + (z * y))) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[((-0.5) / N[(y * t), $MachinePrecision]), $MachinePrecision]}, If[Less[z, -2.8874623088207947e+119], N[(N[(x - N[(t$95$1 / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * N[(N[(2.0 / z), $MachinePrecision] / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Log[N[(1.0 + N[(z * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024294 
(FPCore (x y z t)
  :name "System.Random.MWC.Distributions:truncatedExp from mwc-random-0.13.3.2"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z -288746230882079470000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (- x (/ (/ (- 1/2) (* y t)) (* z z))) (* (/ (- 1/2) (* y t)) (/ (/ 2 z) (* z z)))) (- x (/ (log (+ 1 (* z y))) t))))

  (- x (/ (log (+ (- 1.0 y) (* y (exp z)))) t)))