init-con

Percentage Accurate: 100.0% → 100.0%
Time: 1.5min
Alternatives: 3
Speedup: N/A×

Specification

?
\[es \geq 0 \land es < 1\]
\[1 - es \cdot \left(\sin phi0 \cdot \sin phi0\right) \]
(FPCore (phi0 es)
  :precision binary64
  :pre (and (>= es 0.0) (< es 1.0))
  (- 1.0 (* es (* (sin phi0) (sin phi0)))))
double code(double phi0, double es) {
	return 1.0 - (es * (sin(phi0) * sin(phi0)));
}

real(8) function code(phi0, es)
use fmin_fmax_functions
    real(8), intent (in) :: phi0
    real(8), intent (in) :: es
    code = 1.0d0 - (es * (sin(phi0) * sin(phi0)))
end function

public static double code(double phi0, double es) {
	return 1.0 - (es * (Math.sin(phi0) * Math.sin(phi0)));
}

def code(phi0, es):
	return 1.0 - (es * (math.sin(phi0) * math.sin(phi0)))

function code(phi0, es)
	return Float64(1.0 - Float64(es * Float64(sin(phi0) * sin(phi0))))
end

function tmp = code(phi0, es)
	tmp = 1.0 - (es * (sin(phi0) * sin(phi0)));
end

code[phi0_, es_] := N[(1.0 - N[(es * N[(N[Sin[phi0], $MachinePrecision] * N[Sin[phi0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

f(phi0, es):
	phi0 in [-inf, +inf],
	es in [0, 1]
code: THEORY
BEGIN
f(phi0, es: real): real =
	(1) - (es * ((sin(phi0)) * (sin(phi0))))
END code
1 - es \cdot \left(\sin phi0 \cdot \sin phi0\right)

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Accuracy vs Speed?

Herbie found 3 alternatives:

AlternativeAccuracySpeedup
The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Initial Program: 100.0% accurate, 1.0× speedup?

\[es \geq 0 \land es < 1\]
\[1 - es \cdot \left(\sin phi0 \cdot \sin phi0\right) \]
(FPCore (phi0 es)
  :precision binary64
  :pre (and (>= es 0.0) (< es 1.0))
  (- 1.0 (* es (* (sin phi0) (sin phi0)))))
double code(double phi0, double es) {
	return 1.0 - (es * (sin(phi0) * sin(phi0)));
}

real(8) function code(phi0, es)
use fmin_fmax_functions
    real(8), intent (in) :: phi0
    real(8), intent (in) :: es
    code = 1.0d0 - (es * (sin(phi0) * sin(phi0)))
end function

public static double code(double phi0, double es) {
	return 1.0 - (es * (Math.sin(phi0) * Math.sin(phi0)));
}

def code(phi0, es):
	return 1.0 - (es * (math.sin(phi0) * math.sin(phi0)))

function code(phi0, es)
	return Float64(1.0 - Float64(es * Float64(sin(phi0) * sin(phi0))))
end

function tmp = code(phi0, es)
	tmp = 1.0 - (es * (sin(phi0) * sin(phi0)));
end

code[phi0_, es_] := N[(1.0 - N[(es * N[(N[Sin[phi0], $MachinePrecision] * N[Sin[phi0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

f(phi0, es):
	phi0 in [-inf, +inf],
	es in [0, 1]
code: THEORY
BEGIN
f(phi0, es: real): real =
	(1) - (es * ((sin(phi0)) * (sin(phi0))))
END code
1 - es \cdot \left(\sin phi0 \cdot \sin phi0\right)

Alternative 1: 100.0% accurate, 1.6× speedup?

\[es \geq 0 \land es < 1\]
\[1 - es \cdot \mathsf{fma}\left(-0.5, \cos \left(phi0 + phi0\right), 0.5\right) \]
(FPCore (phi0 es)
  :precision binary64
  :pre (and (>= es 0.0) (< es 1.0))
  (- 1.0 (* es (fma -0.5 (cos (+ phi0 phi0)) 0.5))))
double code(double phi0, double es) {
	return 1.0 - (es * fma(-0.5, cos((phi0 + phi0)), 0.5));
}

function code(phi0, es)
	return Float64(1.0 - Float64(es * fma(-0.5, cos(Float64(phi0 + phi0)), 0.5)))
end

code[phi0_, es_] := N[(1.0 - N[(es * N[(-0.5 * N[Cos[N[(phi0 + phi0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

f(phi0, es):
	phi0 in [-inf, +inf],
	es in [0, 1]
code: THEORY
BEGIN
f(phi0, es: real): real =
	(1) - (es * (((-5e-1) * (cos((phi0 + phi0)))) + (5e-1)))
END code
1 - es \cdot \mathsf{fma}\left(-0.5, \cos \left(phi0 + phi0\right), 0.5\right)
Derivation

  1. Initial program 100.0%

    \[1 - es \cdot \left(\sin phi0 \cdot \sin phi0\right) \]

  2. Taylor expanded in phi0 around inf

    \[\leadsto 1 - es \cdot {\sin phi0}^{2} \]

  4. Applied rewrites100.0%

    \[\leadsto 1 - es \cdot {\sin phi0}^{2} \]

  6. Applied rewrites100.0%

    \[\leadsto 1 - es \cdot \mathsf{fma}\left(-0.5, \cos \left(phi0 + phi0\right), 0.5\right) \]
  7. Add Preprocessing

Alternative 2: 98.9% accurate, 76.8× speedup?

\[es \geq 0 \land es < 1\]
\[1 \]
(FPCore (phi0 es)
  :precision binary64
  :pre (and (>= es 0.0) (< es 1.0))
  1.0)
double code(double phi0, double es) {
	return 1.0;
}

real(8) function code(phi0, es)
use fmin_fmax_functions
    real(8), intent (in) :: phi0
    real(8), intent (in) :: es
    code = 1.0d0
end function

public static double code(double phi0, double es) {
	return 1.0;
}

def code(phi0, es):
	return 1.0

function code(phi0, es)
	return 1.0
end

function tmp = code(phi0, es)
	tmp = 1.0;
end

code[phi0_, es_] := 1.0

f(phi0, es):
	phi0 in [-inf, +inf],
	es in [0, 1]
code: THEORY
BEGIN
f(phi0, es: real): real =
	1
END code
1
Derivation

  1. Initial program 100.0%

    \[1 - es \cdot \left(\sin phi0 \cdot \sin phi0\right) \]

  2. Taylor expanded in phi0 around 0

    \[\leadsto 1 \]

  4. Applied rewrites98.9%

    \[\leadsto 1 \]
  5. Add Preprocessing

Alternative 3: 17.6% accurate, 76.8× speedup?

\[es \geq 0 \land es < 1\]
\[0.3333333333333333 \]
(FPCore (phi0 es)
  :precision binary64
  :pre (and (>= es 0.0) (< es 1.0))
  0.3333333333333333)
double code(double phi0, double es) {
	return 0.3333333333333333;
}

real(8) function code(phi0, es)
use fmin_fmax_functions
    real(8), intent (in) :: phi0
    real(8), intent (in) :: es
    code = 0.3333333333333333d0
end function

public static double code(double phi0, double es) {
	return 0.3333333333333333;
}

def code(phi0, es):
	return 0.3333333333333333

function code(phi0, es)
	return 0.3333333333333333
end

function tmp = code(phi0, es)
	tmp = 0.3333333333333333;
end

code[phi0_, es_] := 0.3333333333333333

f(phi0, es):
	phi0 in [-inf, +inf],
	es in [0, 1]
code: THEORY
BEGIN
f(phi0, es: real): real =
	333333333333333314829616256247390992939472198486328125e-54
END code
0.3333333333333333
Derivation

  1. Initial program 100.0%

    \[1 - es \cdot \left(\sin phi0 \cdot \sin phi0\right) \]

  3. Applied rewrites98.0%

    \[\leadsto 1 - es \cdot 0.5 \]

  5. Applied rewrites98.0%

    \[\leadsto \mathsf{fma}\left(es, -0.5, 1\right) \]

  7. Applied rewrites98.0%

    \[\leadsto \mathsf{fma}\left(-0.5, es, 2\right) - 1 \]

  9. Applied rewrites17.6%

    \[\leadsto 0.3333333333333333 \]
  10. Add Preprocessing

Reproduce

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herbie shell --seed 2026050 +o generate:egglog
(FPCore (phi0 es)
  :name "init-con"
  :precision binary64
  :pre (and (>= es 0.0) (< es 1.0))
  (- 1.0 (* es (* (sin phi0) (sin phi0)))))